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ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 1

Adaptive Step Size Learning with Applications to

Velocity Aided Inertial Navigation System

Barak Or, Member, IEEE, and Itzik Klein, Senior Member, IEEE

Abstract—Autonomous underwater vehicles (AUV) are com-

monly used in many underwater applications. Recently, the usage

of multi-rotor unmanned autonomous vehicles (UAV) for marine

applications is receiving more attention in the literature. Usually,

both platforms employ an inertial navigation system (INS), and

aiding sensors for an accurate navigation solution. In AUV

navigation, Doppler velocity log (DVL) is mainly used to aid

the INS, while for UAVs, it is common to use global navigation

satellite systems (GNSS) receivers. The fusion between the aiding

sensor and the INS requires a deﬁnition of step size parameter in

the estimation process. It is responsible for the solution frequency

update and, eventually, its accuracy. The choice of the step size

poses a tradeoff between computational load and navigation

performance. Generally, the aiding sensors update frequency

is considered much slower compared to the INS operating

frequency (hundreds Hertz). Such high rate is unnecessary for

most platforms, speciﬁcally for low dynamics AUVs. In this work,

a supervised machine learning based adaptive tuning scheme

to select the proper INS step size is proposed. To that end,

a velocity error bound is deﬁned, allowing the INS/DVL or

the INS/GNSS to act in a sub-optimal working conditions, and

yet minimize the computational load. Results from simulations

and ﬁeld experiment show the beneﬁts of using the proposed

approach. In addition, the proposed framework can be applied

to any other fusion scenarios between any type of sensors or

platforms.

Index Terms—Autonomous underwater vehicles, inertial navi-

gation, Kalman ﬁltering, machine learning, step size, supervised

learning, unmanned aerial vehicles.

I. INTRODUCTION

Autonomous vehicles, such as Autonomous underwater

vehicles (AUVs) or multi-rotor unmanned vehicles are com-

monly equipped with an inertial navigation system (INS) and

other sensors [1] to provide real-time information about their

position, velocity, and orientation [2]–[5]. The INS has two

types of inertial sensors, namely, the gyroscopes and ac-

celerometers. The former measures the angular velocity vector,

and the latter measures the speciﬁc force vector. Since the

inertial sensors measurements contain noises, and error terms,

the navigation solution drifts with time. In the underwater

environment, usually, a Doppler velocity log (DVL) is used

to reduce the solution drift with time [6]–[9], while above

the sea surface global navigation satellite systems (GNSS)

measurements are used instead [10], [11].

The inertial sensors, DVL, and GNSS provide discrete in-

formation regarding a vehicle’s continuous motion. Hence,

tracking a vehicle involves a discrete realization of continuous

motion. Such realization requires a step size selection, usually

Submitted: June 2022.

Barak Or and Itzik Klein are with the Hatter Department of Marine

Technologies, Charney School of Marine Science, University of Haifa, Haifa,

3498838, Israel (e-mail: barakorr@gmail.com, kitzik@univ@haifa.ac.il).

made by the designer according to the scenario and compu-

tational constraints [12]–[14]. Moreover, to save power and

extend the sensor/system life, the number of samples received

from each source should be determined such as the information

quality is maintained and the computational load is minimized

[15]. Most of the time, AUV navigate slow underwater for

a long time. Thus, there is no need to obtain a navigation

solution in a high frequency, except in situation of maneuvers

where the drift might grow and there is a need for a momentary

high computational load [3]. To cope with this trade-off an

adaptive step size may be used.

In [16], an adaptive scheme for the step size, based only

on the vehicle speed, was suggested for dealing with sensor

scheduling in target tracking scenarios. An adaptive scheme

was also suggested in [17], where the step size is based

only on the vehicle’s distance to the target. It improves the

energy efﬁciency during target tracking scenarios. In [12],

a simple criterion was suggested to deﬁne the step size for

sensor measurements to minimize computational load and still

provide moderate navigation performance. This approach is

based on the predictor and corrector of the linear discrete

Kalman Filter (KF) [18], [19], where the main idea is to keep

the discretized implementation of the continuous process with

a lower numerical error. Later, an adaptive scheme to update

the step size in real-time scenarios, with varying discrete noisy

measurements was presented for constant velocity (CV) and

constant acceleration (CA) models [20].

Focusing on inertial measurements step size, in a sensor fusion

scenario like INS/DVL or INS/GNSS, the fusion is carried out

using a nonlinear ﬁlter such as the Extended KF (EKF). There,

inertial measurements are used in the system model while the

aiding measurements are used in the ﬁlter measurement model

to update the navigation state. The inertial sensors operate in

a much faster frequency (tens or hundreds of Hertz) than the

aiding sensors (several Hertz). As a consequence, approaches

like in [12], [15]–[17], are not suitable for such setups, as they

assume constant step sizes.

In order to avoid the need of constant step size, recent works

explore the possibilities of using classical machine learning

(ML) or deep learning (DL) based approaches [21]. Such

approaches recently lead to state of the art performance in

various ﬁelds such as computer vision [22] and natural lan-

guage processing [23]. Recently, ML and DL approaches are

adopted in navigation applications. Among them are pedestrian

dead reckoning where DL approaches are used to regress the

user change in distance and heading [24]–[28]. In addition,

DL based networks were used for attitude estimation using

inertial sensors [29]–[31] . ML approaches showed also better

performance in stationary coarse alignment both in accuracy

arXiv:2206.13428v1 [cs.RO] 27 Jun 2022

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 2

and time to converge [32]. This trend of integrating ML ap-

proaches with classical INS applications raises the motivation

to adapt such approaches also in the described problem of

ﬁnding an adaptive step size.

In this area, a recent work by Dias et al. [15], discusses an

adaptive step size of sensor networks, where online reinforce-

ment learning technique is adopted to minimize the number

of transmissions of the reported data. They showed a high

reduction of energy while keeping the average information

quality. However, that approach is limited to large step sizes

and focused on a slow dynamics system, which is less relevant

for navigation applications.

In this paper, a typical scenario of an adaptive step size

determination for high rate inertial measurement unit (IMU)

aided by a low-rate sensor such as DVL or GNSS is consid-

ered. There, the quality/amount-of-measurements trade-off to

minimize velocity error as a function of the IMU step size is

addressed in the EKF framework. In the proposed approach,

ML models are used to predict the sub-optimal IMU step size,

and handle the non-linearity of the INS model. Establishing a

relationship between navigation features and their sub-optimal

IMU step sizes can be applied in a real-time setting to solve

the IMU step size conﬂict (accuracy vs. computational load).

The main contributions of this paper are:

1) A numerical study of the effect of the inertial sensor step

size on the velocity error in INS/DVL and INS/GNSS

typical fusion scenarios.

2) Derivation of a learning-based scheme to determine an

adaptive IMU step size as a function of the velocity error.

3) Online integration of the proposed learning scheme with

EKF implementation for the navigation ﬁlter.

To validate the proposed approach two numerical examples

of an AUV and a quadrotor are addressed, as well as quad-

copter ﬁeld experiments. Both simulations and experiments re-

sults show the beneﬁts of implementing the proposed learning-

based approach.

The rest of the paper is organized as follows: Section II deals

with the problem formulation for INS/DVL and INS/GNSS

models. Sections III presents the importance of step size

selection, followed by novel learning-based step size tuning

approach, where the feature engineering, database generation

process, and adaptive tuning scheme with the INS are dis-

cussed. Section IV presents the simulations and ﬁeld experi-

ment results, and Section V gives the conclusions.

II. ADA PT IV E NAVIG ATIO N FILTER

The nonlinear nature of the INS equations requires a non-

linear ﬁlter. The most common ﬁlter for fusing INS with

external aiding sensors is the es-EKF [1]. There, the errors

are estimated and subtracted from the state vector. When

considering velocity aided INS, the position vector is not

observable [7]. Hence, it is not included in the error-state

vector, deﬁned as:

δx=δvnδεnbabgT∈R12×1,(1)

where δvn∈R3×1is the velocity vector error states expressed

in the navigation frame, δεn∈R3×1is the misalignment

vector expressed in the navigation frame, ba∈R3×1is

the accelerometer bias residuals vector expressed in the body

frame, and bg∈R3×1is the gyro bias residuals vector

expressed in the body frame. The navigation frame is denoted

by nand the body frame is denoted by b. The navigation frame

center is located at the body center of mass, where the x-axis

points to the geodetic north, z-axis points down parallel to

local vertical, and y-axis completes a right-handed orthogonal

frame. The body frame center is located at the center of mass,

x-axis is parallel to the longitudinal axis of symmetry of the

vehicle pointing forwards, the y-axis points right, and the z-

axis points down such that it forms a right-handed orthogonal

frame. The linearized, error-state, continuous-time model is

δ˙x =Fδx+Gδw,(2)

where F∈R12×12 is the system matrix, G∈R12×12 is

the shaping matrix, and δw=wawgwab wgb T∈

R12×1is the system noise vector consisting of the accelerom-

eter, gyro, and their biases random walk noises, respectively

[33]. The system matrix, Fand the shaping matrix Gare

provided in the appendix. We deﬁne Tn

bas the transformation

matrix between body frame and navigation frame. The corre-

sponding discrete version of the navigation model (for small

step sizes), as given in (2), is

δxk+1 =Φkδxk+Gkδwk.(3)

The transition matrix, Φk, is deﬁned by a ﬁrst order approxi-

mation as

Φk

∆

=I+F∆t, (4)

kis a time index, δwkis a zero mean white Gaussian noise,

and Iis an identity matrix.

The step size for the INS calculations is deﬁned by

∆tk

∆

=tk−tk−1,(5)

where each step size is related to the IMU frequency, νI MU ,

by

∆tk=1

νIM U

.(6)

The discretized process noise is given by

Qd

k=GQcGT∆tk,(7)

where Qcis the continuous process noise matrix.

The discrete es-EKF is used to fuse the INS with external

measurements. The initial error state and error state covariance

are deﬁned as [1], [34]

δˆx0=012×1

P0=Qd,(8)

where δˆx0is the initial estimate error-state vector, and P0is

the initial covariance error.

The error-state vector is initialized every iteration, as following

δˆx−

k=0.(9)

The error covariance propagation (prediction) is given by

P−

k=Φk−1Pk−1Φk−1T+Qd

k−1,(10)

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 3

where Pk−1is the estimate from previous state, k−1. The

measurement arrives at time j, and then ﬁlter update is made.

The Kalman gain is given by

Kj=P−

jHjThHjP−

jHjT+Rd

ji−1,(11)

where Rdis the discrete measurement noise covariance,

assumed to be constant.

The error-state estimate update is given by

δˆxj=Kjδzj,(12)

where

δzj=ˆzj−zj(13)

is the measurement residual vector, deﬁned as the difference

between the estimated (ˆzj) and the actual (zj) measurements.

Finally, the error covariance update (correction) is given by

Pj= [I−KjHj]P−

j.(14)

A. Velocity measurement models

Two types of velocity measurement models are considered:

1) DVL and 2) GNSS. Regardless of the measurement model,

the velocity measurements are available in a constant fre-

quency, with a different step size from the IMU. The step

size of the aiding velocity sensor is given by

∆τj=τj−τj−1, j = 1,2, ... (15)

and related to the aiding sensor sampling frequency, νAiding ,

by

∆τj=1

νAiding

.(16)

Without the loss of generality, it is assumed a constant step size

for the aiding sensor measurements. Hence, the subscript jis

omitted for ∆τ. Commonly, the IMU has a higher frequency

rate than the aiding sensor, thus, the following assumption is

made:

∆τ∆tk,∀k. (17)

1) DVL measurement model: After processing, DVL out-

puts the AUV velocity vector in the DVL frame, vd

DV L. Then,

it is transformed to the body frame, vb

DV L, and eventually to

the navigation frame, vn

DV L, where it is used in the navigation

ﬁlter. Thus,

vn

DV L =Tn

bTb

dvd

DV L,(18)

where Tb

dis the transformation matrix from the DVL frame

to the body frame. For simplicity, it is assumed that Tb

dis

accurately known, and therefore, removed in further analysis.

Linearizing (18) yields [35]

δvb

DV L =Tb

nδvn−Tb

n(vn×)δεn.(19)

The corresponding measurement residual is given by

δzb

DV L,j =HDV L,j δxj+ςDV L,j,(20)

where ςDV L,j ∼ N 0,Rd

DV L∈R3×1is an additive

discretized zero mean white Gaussian noise. It is assumed that

ςjand δwjare uncorrelated. The corresponding time-variant

DVL measurement matrix is given by

HDV L,j =hTb

nj−Tb

nj(vn×)j03×6i∈R3×12.(21)

2) GNSS velocity measurement model: The GNSS receiver

outputs the velocity vector in the navigation frame. Hence,

the corresponding time-invariant GNSS measurement matrix

is given by

HGNS S =I3×303×9∈R3×12.(22)

The corresponding measurement residual is given by

δzb

GNS S ,j =HGN SS δxj+ςGN SS ,j ,(23)

where ςGNS S ,j ∼ N 0,RdGNS S ∈R3×1is an additive

discretized zero mean white Gaussian noise. It is assumed that

ςjand δwjare uncorrelated.

III. ADAP TI VE S TE P SI ZE L EA RN IN G

A. Motivation: the importance of step size selection

To demonstrate the inﬂuence of the step size on vehicle’s

velocity error, a simpliﬁed simulated vehicle trajectory, shown

in Figure 1, was used. The simulation parameters are summa-

rized in Table I. As the example was conducted for a short

period (T= 240[s]), the IMU error model was simpliﬁed to

include only zero mean white Gaussian noise:

¯

fb=fimu

b+wa,(24)

and

¯ωib =ωimu

ib +wg,(25)

where fimu

band ωimu

ib are true simulated outputs of the

accelerometer and gyroscope, respectively.

In order to evaluate the navigation performance, Monte Carlo

(MC) simulation with 100 iterations was made. The av-

eraged velocity root mean squared error (RMSE) for the

entire scenario is 0.06[m/s]which was obtained by setting

∆t= 0.01[s].

TABLE I

INS/GNSS SI MUL ATIO N PARA MET ER S

Description Symbol Value

GNSS velocity noise (var) R11, R22 , R33 0.0042[m/s]2

GNSS step size ∆τ1[s]

Accelerometer noise (var) Q11 , Q22, Q33 0.022[m/s2]2

Gyroscope noise (var) Q44, Q55 , Q66 0.0022[rad/s]2

IMU step size ∆t0.01[s]

Num. Monte Carlo iterations N100

Simulation duration T240[s]

Initial velocity vn

0[5,0,0]T[m/s]

Initial position pn

0[320,340,5[m]]T

As the velocity accuracy is affected by the predetermined

step size of the IMU, a sub-optimal step size satisﬁes the

following condition:

∆t∗= arg min

∆t∈T

[E(∆t)− B],B>0,(26)

where

E(∆t)∆

=Ekδvn

T rue (∆t)k2.(27)

The argument ∆t∈ T ⊂ [∆tmin,∆tmax]minimizes the

difference between the 2nd Euclidean norm of the mean

averaged velocity (speed) error vector (E) and a design value,

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 4

Fig. 1. Estimated and ground truth simulated trajectories. Blue line shows

ground truth and the black doted line shows the estimated trajectory. The plots

are presented in a local Cartesian coordinate system.

B. Thus, the criterion allows velocity (speed) error up to B

for the sub-optimal step size. As Bis a design parameter, one

can choose it according to the platform and scenario at hand.

In this work, we allow an averaged velocity error of 0.1[m/s].

Obviously, when Bgoes to zero, ∆t→0, and as a conse-

quence the computational load increases. Hence, to ﬁnd a trade

off between accuracy and computational load, the condition,

B>0, must be satisﬁed. In due course, we performed 100

MC simulations each with 10 different IMU step sizes for

three cases (different GNSS step size, ∆τ):

∆t∈˜

T=0.002,0.004,0.008,0.01,0.016

0.02,0.032,0.04,0.05,0.1.(28)

The results are summarized in Figure 2, where each point

represents the averaged velocity (speed) RMSE. For all

scenarios, as the step size increases, the averaged velocity

(speed) RMSE also increases. As seen in the ﬁgure, the

change of the averaged velocity (speed) RMSE slowly

converge to steady-state values for all step sizes per scenario.

Hence, there is a ∆t, which is too large to carry the navigation

information, and it is associated with a high error. On the other

hand, given a bound for the averaged velocity RMSE, B, a

moderate value of ∆tcan be deﬁned where the computational

load will be minimized. For example, if ∆t= 0.05[s]or

∆t= 0.1[s]for case [2] (Figure 2), are chosen, the same

averaged velocity (speed) RMSE of 0.01[m/s]is achieved.

For this example, 50% of the computational load can be

reduced without affecting the velocity error accuracy. Other

cases consider the GNSS step size (∆τ) as very small value

(not necessarily available in the market, yet) - to demonstrate

the impact of high-rate update.

B. Supervised learning formulation

The power of ML rises the ability to solve many difﬁcult and

non-conventional tasks. To determine the step size, the prob-

lem is formulated in a Supervised Learning (SL) approach; A

Fig. 2. Velocity sensitivity to step size for ﬁve different scenarios with various

GNSS step size (∆τ). Complete trajectory and variances values are provided

in Table 1.

feature set is deﬁned where kinematic and statistics measures

are considered. Formally, we search for a model to relate an

instance space, X, and a label space, Y. We assume that there

exists a target function, F, such that Y=F(X). Generally,

the SL task is to ﬁnd F, given a ﬁnite set of labeled instances:

{Xk,∆t∗

k}N

k=1 .(29)

The SL aims to ﬁnd a function ˜

Fthat best estimates F. A

loss function, l, is deﬁned to quantify the quality of ˜

Fwith

respect to F. The overall loss is given by

LY,ˆ

Y(X)∆

=1

M

M

X

m=1

l(y, ˆy)m,(30)

where Mis the number of examples, and min the example

index. Minimizing Lin a training/test procedure leads to the

target function. The step size tuning problem is formulated as a

classiﬁcation problem, where only two classes are considered:

Y={0.04,0.002} ∈ R1×2.(31)

Ideally we would like to minimize the computational cost

without inﬂuencing the accuracy. Yet, due to the inherit

tradeoff this is not possible. Therefore, we would like to

minimize the computational cost with resulting minimum

accuracy degradation. As a consequence, only the step-size

is considered in the loss function, l, given by

lm

∆

=∆t∗

m−d

∆tm2,(32)

where d

∆tmis the estimated step size value obtained by the

learning model during the training process. The main reasons

for considering this problem as a classiﬁcation task and not

regression task are:

1) Filter robustness: Minimizing the amount of step size

switching along the navigation process.

2) ML model robustness: As there are only two classes in

the label space. By doing so, the deterministic label space

avoid invalid values and improve real-time performance.

Notice, that in (30) two different step sizes are considered. Yet,

if needed, the proposed framework can be applied for more

different values pending on the scenario at hand. The major

beneﬁt of deﬁning the problem as a bi-classiﬁcation predictor,

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 5

is that we minimize the number of “chattering” between many

step size values (might lead to unstable ﬁlter). Also, using two

values, one big 0.04[s]and the other small 0.002[s], presents

clearly the computational effort reduction. Later, an example

supporting the bi-classiﬁcation choice is provided.

C. Feature engineering

Sixteen high-level and low-level features, are considered:

X=Xhigh ,Xlow ∈R1×16,(33)

where

1) High-level features: A group of features that contains

physical values of various ﬁlter and vehicle parameters in

the scenario. Mean and square root are commonly used

in many types of classiﬁcation/regression problems and

thus used:

Xhigh =(qQg11 ,pQa11 ,qRd

11 ,

∆τ, E(ˆvn)2,Eˆϕ2,Eˆ

θ2,Eˆ

ψ2)∈R1×10,

(34)

where the subscript 11 stands for the ﬁrst element of a

matrix, and Eis the expected value operator, calculated

based on a moving average of the last 50 values. E(ˆvn)2

contains three features for north, east, and down velocity

components. φ, θ, and ψare the body angles (brieﬂy

explained in the appendix).

2) Low-level features: The low-level features are scalar

values, created based on combination and modiﬁcation

of the high-level features, as summarized in Table II.

Low-level features were chosen due to the dynamic body

behavior and noise characteristics.

Generally, the designer can chose additional/different features

for the classiﬁcation task.

TABLE II

LOW-LE VEL F EATU RES

Index Feature Description

Xlow

1qQω11 +Qfb11 +Rd

11 Noise StD 2-norm

Xlow

2r1

Qω11

+1

Qfb11

+1

Rd

11

Noise ’mean’

Xlow

3

√Qω11 +qQfb11

qRd

11

IMU + aiding sensor noise ratio

Xlow

4Rd

11 ∆τAiding sensor noise variance

Xlow

5kˆvnk2Vehicle speed

Xlow

6

√Qω11 +qQfb11

qRd

11 kˆvnk2Xlow

3∗ X low

5

D. Database generation process

As a preliminary stage of training, a dataset should be

generated. Then, it could be processed into the ML model.

A velocity error, B, was deﬁned using different trajectories.

There, the vehicle traveled along them several times with

various step sizes to ﬁnd and store those that minimized

vehicle velocity vector (speed) error. The various trajectories

were created by modifying the radius of a circular motion,

straight lines, and general curves. The process of generating

such trajectories is divided into two parts: INS simulation with

perfect IMU measurements to produce the GT trajectories

and store them, and noisy velocity aided INS simulation in

order to create noisy examples with their corresponding ∆t∗

satisfying a desired bound of velocity error, B. The IMU noise

variance values as also velocity aiding sensor noise values are

summarized in Table III.

TABLE III

VALUE RANGES FOR DATABA SE GE NE RATI ON

Description Value Range Amount

Aiding velocity noise (var) Rii ∈[0.0001,0.1]2[m/s]210

Aiding velocity step size ∆τ∈[0.1,2] [s] 6

Accelerometer noise (var) Qii ∈[0.0005,0.5]2m/s2210

Gyroscope noise (var) Qii ∈[0.0001,0.1]2[rad/s]210

The database generation process is described in Figure

3. IMU noise variances were ﬁrstly set, similarly to the

simpliﬁed IMU error model provided in (24) −(25) with

different noise variances values. Then, IMU readings and

aiding sensor measurements (constant step size) are processed

into the velocity aided INS scheme and provide the vehicle

navigation solution and their state errors as well. As the focus

is to determine a sub-optimal ∆tto minimize velocity error,

the error upper bound, B, was chosen to be 0.1[m/s](other

values can be examined instead). If the condition (26) is

satisﬁed, the example is stored. Else, the step size of the IMU,

∆t, is reduced. Repeating this process for many scenarios

yields a large dataset, enabling model training. Therefore, the

examples in the dataset have the smallest step size, within

the deﬁned error bound, achieving the objective of trade-off

balance between computation and accuracy

Fig. 3. Database generation process. IMU and accurate velocity measurements

enter the velocity aided INS, then the velocity error is calculated. Given the

GT velocity, the system decides if step-size should be reduced or not.

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 6

E. Adaptive tuning scheme

Applying the suggested tuning approach in online setting

involves integration of the velocity aided INS with the

classiﬁer, as presented in Figure 4. Algorithm 1 gives the

velocity aided INS with adaptive step size tuning algorithm.

Algorithm 1 Velocity aided INS with adaptive step size tuning

Input: ωib,fb,vAiding ,∆t0,∆τ, T, tuning Rate

Output: vn, εn

Initialization :vn

0, εn

0

LOOP Process

1: for t= 0 to Tdo

2: obtain ωib,fb

3: solve navigation equations (3)

4: if (mod(t, ∆τ)=0)then

5: obtain vAiding (20),(23)

6: update navigation state using the es-EKF (8)-(14)

7: end if

8: Calculate features and predict ∆t∗

k+1.

9: if mod(t, tuningRate)=0then

10: calculate X(33)-(34)

11: ∆t∗

k+1 =˜

Ftrained (X)

12: end if

13: end for

Fig. 4. Step size adaptive tuning by applying the ML classiﬁer. The features

(high level and low level) are calculated based on temporal data. Then, they

are processed into a classiﬁer that outputs the sub-optimal step size.

IV. RES ULT S

A. Classiﬁcation methods comparison

In order to ﬁnd the most suitable SL prediction model,

various classiﬁcation models have been explored: ﬁne tree

(decision tree with many leaves), Naive Bayes, K-Nearest

Neighborhood (KNN), Support Vector Machine (SVM), Lo-

gistic regression, and Ensemble (boosted trees) [21], [36]. The

comparison process was made for 5 cases, with different IMU

variance values, using a database (created as described in

Section III.D) consisting of 18,000 examples. Those includes

motion along straight lines with various velocities, and circles

with different radiuses and velocities. All classiﬁers were bi-

classiﬁers, with small and large step sizes for better model

robustness (brieﬂy explained in the appendix). The IMU, and

aiding sensors noise covariances were tuned with different

values to enrich the dataset. Note, that this dataset was created

to handle with both INS/DVL and INS/GNSS fusion scenarios.

Two training paradigms were considered: train/test with 80/20

ratio, and cross validation with ﬁve folds. The vehicle’s

dynamics was set by tuning the IMU and selecting the initial

kinematic conditions.

To evaluate the proposed models performance, the area under

curve (AuC) measure was employed (see appendix for further

explanations). For that, the positive value was deﬁned as

P∆t= 0.002[s], and the negative value as N∆t= 0.04[s]. The

second criterion used in order to evaluate the proposed models

is the accuracy measure. Both criterions were calculated for

each of the candidate models.

Classiﬁcation results comparing ML approaches are provided

in Figure 5. Each approach achieves AuC score and accuracy

score for their classiﬁcation performance, where once it was

made by train/test paradigm and once by cross-validation

paradigm. All models obtained more than 0.88 accuracy and

AuC rates for both paradigms. The SVM obtained high ac-

curacy rate (0.95) using the train/test paradigm, and high rate

in the cross validation paradigm (also, a good performance

according to the AuC rate). The Ensemble method slightly

outperforms the SVM approach according to the AuC (both

train/test and cross validation) as well as according to accuracy

(cross validation). As the accuracy of both methods is similar,

the SVM was chosen as, in general, it is known to be a

robust classiﬁer, as it maximizes the hyperplane margins [37].

Another justiﬁcation to consider the SVM classiﬁer is the

trained model computational time. To that end, the excitation

time was measured in the algorithm working environment (In-

tel i7-6700HQ CPU@2.6GHz 16GB RAM with MATLAB).

The Ensemble model averaged iteration calculation time was

0.015[s] while the SVM was 0.001[s]. Hence, the SVM is

15 times faster than the Ensemble, which is a very important

property in real time applications. These reasons lead us to

choose the SVM as the optimal classiﬁer for this task, as we

deal with real-time scenarios and aim to keep the navigation

ﬁlter robustness and efﬁcient. The resulted ROC curve with

a chosen classiﬁer, (FPR = 0.03, T P R = 0.87), received

AuC of 0.98, with accuracy of 0.95 (obtained by a train/test

paradigm). Hence, the SVM classiﬁer was chosen for further

analysis.

Fig. 5. Learning model comparison with AuC and accuracy performance

measures.

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 7

B. MRMR based feature rank

The learning-based models were trained using 16 features.

In many real-time application the computational time of these

features is critical and might take long time and eventually

result in system latency or memory constraints. In order

to avoid that, feature dimensionality reduction methods are

applied. There, the features rank approach is used to select

the most contributing features. One of the classical and com-

mon feature ranking approaches is the minimum redundancy

maximum relevance (MRMR) method [38]. It was used to

rank the feature set (32). Figure 6 summarizes the results

showing that Eˆ

ψ2,pQa11 , and pRd

11are the three most

important features. Although all features were used in the

current research, the MRMR showed that the high level

features contribute more to the classiﬁcation process. Thus,

if some computational constraint is required they could be

removed from the analysis/model.

Fig. 6. Feature importance ranking as obtained by the MRMR method.

C. Simulations

Two different simulation scenarios to validate the velocity

aided INS with adaptive step size tuning were made. First, an

INS/GNSS with scenario parameters given in Table IV, and

secondly, an INS/DVL with scenario parameters given in Table

V. Both simulated trajectories are constructed by lines and

curves, that the ML classiﬁer was not trained on. However, it is

expected the model will successfully capture the dynamics and

statistics along the trajectories to predicting the sub-optimal

∆t∗, as it was trained over 18,000 examples (Section III.D)

with various scenarios. For better comparison, we add one

classical approach for step size tuning as a function of the

velocity, given by [16]:

∆t∗

k+1 (kvnk2) = ∆tmin (kvnk2)k> vT resh

∆tmax (kvnk2)k≤vT resh ,(35)

where the velocity threshold, vT resh is determined by the

designer, upon the real-time scenario. In the adaptive setting,

once the ML classiﬁer predicts a different step size from the

one it used in the last 20 steps, the algorithm tunes the updated

step size for the next iteration.

For the trajectory shown in Figure 1, graphs of the predicted

∆tas a function of time according to the ML classiﬁer and the

classical approach (35) are plotted in Figure 7, for INS/GNSS

simulation. There, the ML classiﬁer predicted mostly ∆t∗=

0.04[s], except for short time interval ([50,58][s]), where it

predicted ∆t∗= 0.002[s]. The velocity error results are shown

in Table VI, where, in addition to the adaptive tuning, two

constant step sizes were used for comparison. Applying the

small step size (∆t= 0.002[s]) results in mean velocity

error of 0.145[m/s], which is the lowest error associated with

high computational load of 120,000 iterations, and maximum

velocity error of 0.41[m/s]. From the other side, applying

the larger step size (∆t= 0.04[s]), used only 6,000 iterations

(only 5% of the smaller step size) results in mean velocity error

of 0.187[m/s](less than 0.05[m/s]increase) and maximum

velocity error of 0.655[m/s]. By applying the adaptive step

size tuning approach, a mean velocity error of 0.181[m/s]

was obtained, with only 9,381 iterations. This is less than 10%

of the conservative approach with IMU step size of 0.002[s],

and also yields a lower velocity error than the large step size.

The maximum velocity error with the adaptive step size is

0.37[m/s], lower than both constant cases.

In Figure 8, the INS/DVL simulated trajectory is shown. The

changes of ∆t∗during time according to the ML classiﬁer

and the classical approach (35) are plotted in Figure 9. There,

the ML classiﬁer predicted ∆t∗= 0.002[s]for the ﬁrst 17[s]

of the trajectory, and then predicted ∆t∗= 0.04[s]until the

end of the trajectory. The velocity error results are shown in

Table VII, where, similarly to the INS/GNSS simulation, two

constant step sizes were used to compare the adaptive tuning

step size. Applying the smaller step size (∆t= 0.002[s])

results in 0.015[m/s]mean velocity error and maximum

velocity error of 0.019[m/s], which is the error associated with

high computational load of 20,000 iterations. From the other

side, applying the larger step size (∆t= 0.04[s]), used only

1,000 iterations, (only 5% of the smaller step size) results

in 0.0216[m/s]mean velocity error (less than 0.07[m/s]

increase) and maximum velocity error of 0.046[m/s]. By ap-

plying the adaptive step size tuning approach, a mean velocity

error of 0.012[m/s]was obtained with only 9,360 iterations.

This is less than a half from the conservative approach with

IMU step size of 0.002[s], and also yields a lower mean

velocity error. A maximum velocity error of 0.028[m/s]were

obtained. For both INS/GNSS and INS/DVL simulations, the

classical method for determining ∆tkbased on the vehicle

speed, (35), obtained insufﬁcient computational load with

higher mean velocity error, where for the INS/GNSS a mean

velocity error of 0.203[m/s]was obtained with over than

70,000 iterations. and for the INS/DVL a mean velocity error

of 0.280[m/s]was obtained with nearly 10,000 iterations. The

threshold was determined as the initial vehicles speed in both

scenarios. To summarize, while using our proposed adaptive

approach, the average velocity error has increased by only

0.077[m/s]While using only about 1/7of the computational

load.

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 8

TABLE IV

INS/GNSS WITH ADAPTIVE STEP SIZE SIMULATION PARAMETERS.

Description Symbol Value

GNSS velocity noise (var) R11, R22 , R33 0.02[m/s]2

GNSS step size ∆τ1[s]

Accelerometer noise (var) Q11 , Q22, Q33 0.042[m/s2]2

Gyroscope noise (var)) Q44, Q55 , Q66 0.0032[rad/s]2

Simulation duration T240[s]

Initial velocity vn

0[5,0,0]T[m/s]

Initial position pn

0[320,340,5[m]]T

Velocity threshold vT resh 5[m/s]

TABLE V

INS/DVL WITH ADAPTIVE STEP SIZE SIMULATION PARAMETERS.

Description Symbol Value

DVL noise (var) R11, R22 , R33 0.004[m/s]2

DVL step size ∆τ1[s]

Accelerometer noise (var) Q11, Q22, Q33 0.022[m/s2]2

Gyroscope noise (var) Q44 , Q55, Q66 0.0022[rad/s]2

Simulation duration T40[s]

Initial velocity vn

0[1,0,0]T[m/s]

Initial position pn

0[320,340,−5[m]]T

Velocity threshold vTresh 1[m/s]

TABLE VI

INS/GNSS SIMULATION RESULTS.

∆t[s]Mean δvn[m/s]Max δvn[m/s]Iterations

Adaptive (ours) 0.181 0.370 9,381

0.002 0.145 0.410 120,000

0.04 0.187 0.655 6,000

∆tkvnk20.203 0.204 70,410

Fig. 7. INS/GNSS simulation with sub-optimal step size, ∆t∗, based on the

ML classiﬁer and classical ∆tkvnk2as a function of time for a duration

of 4 minutes.

TABLE VII

INS/DVL SIMULATION RESULTS.

∆t[s]Mean δvn[m/s]Max δvn[m/s]Iterations

Adaptive (ours) 0.012 0.028 9,360

0.002 0.015 0.019 20,000

0.04 0.0216 0.046 1,000

∆tkvnk20.0283 0.076 9,884

Fig. 8. INS/DVL simulation trajectory for an AUV. The vehicle moves at the

same sea level (−5[m]) and performs a rectangular motion. The blue line is

for the GT trajectory and the dotted black line is for the estimated trajectory.

Fig. 9. INS/DVL simulation with sub-optimal step size, ∆t∗, based on the

ML classiﬁer and classical ∆tkvnk2as a function of time for a duration

of 40 seconds.

D. Field experiment

A ﬁeld experiment using a quadrotor was performed (Figure

10). The altitude of the quadrotor was kept constant and

the horizontal trajectory is shown in Figure 11. The 12

error state model, described in Section II.A, was used to

obtain the navigation solution. The ﬁlter was updated by

velocity measurements from a GNSS receiver and the resulting

navigation solution was compared with a GT measurements,

obtained using an RTK device. To examine different (from the

ones used in our simulations) and challenging scenarios (for

the proposed ML algorithm), an “8-ﬁgure” shape trajectory

was applied, to include accelerations/ declarations, as also

turns for part of the time, and almost “straight” lines for the

rest. Also, different parameters values were examined in the

ﬁeld experiment resulting in more cases that were examined

and covered for better conclusion and generalization of the

ML approach, keeping the ML strategy the same. Experiment

parameters are provided in Table VIII. In Algorithm 1, line

11, for the experiment, the optimal step size is 0.02[s]. There

is no need for additional training as the experimental dataset

was addressed as a new test dataset, that is the ML was trained

on the simulation training dataset.

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 9

The experiment error results using the classical approach,

∆tk= 0.002[s],∆tk= 0.02[s], and ∆t∗from the sug-

gested adaptive tuning approach are summarized in Table IX.

It appears that 1800 iterations results in 0.128[m/s]mean

velocity error, and increasing the number of iterations to

18,000 yields a meaningful reduction where only 0.01[m/s]

mean velocity error is obtained. A maximum velocity error of

6.25[m/s]was obtained for a fractional initialization moment,

both for the adaptive step size and the constant step size of

∆tk= 0.002[s]. Our suggested adaptive tuning algorithm

founds a sub-optimal solution, where only 9,900 iterations

yields a 0.02[m/s]mean velocity error (as deﬁned by setting

B). The designer controls the amount of iterations according

to velocity RMSE criterion. In this experiment, the number

of iteration using ∆tk= 0.02[s]increases adaptive by a

factor of almost 6 (from 1800 to 9900) to meet designer’s

criterion. From the other side, applying the adaptive tuning

scheme results in 45% reduction of number of iterations

obtained while setting ∆tk= 0.002[s]with only 0.01[m/s]

increase of mean velocity error. The maximum velocity error

is 5.82[m/s]for a fractional initialization moment. In this

experiment we obtained a nearly linear relationship between

the velocity mean error and the step size. This result conﬁrms

our numerical simulation from III.A. The classical method for

determining ∆tkbased on the vehicle speed, (35), obtained

insufﬁcient computational load with higher mean velocity

error: a mean velocity error of 0.037[m/s]was obtained with

4,500 iterations. The changes of ∆t∗during time according

to the ML classiﬁer and the classical approach (35) are plotted

in Figure 12.

Fig. 10. DJI matrice 300 in a ﬁeld experiment.

V. CONCLUSIONS

A proper choice of the step size is important for imple-

menting velocity aided INS. The step size depends on various

navigation parameters. In real life, these parameters and their

behavior in the dynamic environment are partly known. A

novel ML-based scheme to adaptively tune an appropriate step

size, together with the es-EKF implementation was proposed.

According to this scheme, the designer should set an averaged

Fig. 11. Field experiment Quadrotor GT trajectory and its estimated one.

TABLE VIII

EXP ERI ME NT PAR AME TER S

Description Symbol Value

GNSS vel. noise (var) - horizon. R11, R22 0.003[m/s]2

GNSS vel. noise (var) - vertical R33 0.005[m/s]2

GNSS step size ∆τ0.1[s]

Accelerometer noise (MPU-9250) Q∗

11, Q∗

22, Q∗

33 300[µg/√Hz]

Gyroscope rate noise (MPU-9250) Q∗

44, Q∗

55, Q∗

66 0.01[dps/√Hz]

Experiment duration T35[s]

Initial velocity vn

0[0,0,0]T[m/s]

Initial position pn

0[32.10,34.80,0]T

Accelerometer bias noise Q∗

7, Q∗

8, Q∗

9[1,1,1][m/s2]2

Gyroscope bias noise Q∗

10, Q∗

11, Q∗

12 [1,1,1][rad/s]2

velocity threshold vT resh 7[m/s]

TABLE IX

ERRO RS AS A F UN CTI ON O F STE P SI ZE

AND THE RESULTING NUMBER OF ITERATIONS

∆tk[s] Mean δvn[m/s]Max δvn[m/s]Iterations

Adaptive (ours) 0.0217 6.25 9900

0.002 0.0111 6.25 18000

0.02 0.1280 5.82 1800

∆tkvnk20.037 5.82 4,500

Fig. 12. INS/GNSS experiment with sub-optimal step size, ∆t∗, based on the

ML classiﬁer and classical ∆tkvnk2as a function of time for a duration

of 35 seconds.

velocity error bound, where the ML classiﬁer predicts the sub-

optimal step size to provide the navigation solution without

exceeding this bound, in real-time scenarios. Extensive simula-

tions and a ﬁeld experiment demonstrated the efﬁciency of this

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 10

methodology in commonly used vehicle tracking problems.

The proposed scheme minimized the computational load with

minimum inﬂuence on velocity estimate error. The scheme

was validated using two simulations and ﬁeld experiment.

There, the relationship between the velocity RMSE and the

IMU step size was presented. In the INS/DVL, INS/GNSS

and ﬁeld experiment, we measured the velocity only and

found that we can use lower number of iterations, and by

that minimize computational load, with a sufﬁcient velocity

RMSE by applying the suggested scheme. In this work, for

demonstrative proposes, only two different step size were

examined, yet the proposed approach can be easily elaborated

to more different sizes. In addition, the goal of this work was

focused on velocity aided INS, however the proposed approach

can be used with any other aiding sensors and to any other

platform.

ACK NOW LE DG EM EN T

BO was supported by The Maurice Hatter Foundation.

APPENDIX

A. INS Equations of Motion

The INS equations of motion include the rate of change

of the position, velocity, and the transformation between the

navigation and body frame, as shown in Fig.12.

The position vector is given by

pn=φλhT∈R3×1,(36)

where φis the latitude, λis the longitude, and his the altitude.

The velocity vector is Earth referenced and expressed in the

North-East-Down (NED) coordinate system:

vn=vNvEvDT∈R3×1,(37)

where vN, vE, vDdenote the velocity vector components in

north, east, and down directions, respectively. The rate of

change of the position is given by [1]

˙pn=

˙

φ

˙

λ

˙

h

=

vN

RM+h

vE

cos(φ)(RN+h)

−vD

,(38)

where RMand RNare the meridian radius and the normal

radius of curvature, respectively. The rate of change of the

velocity vector is given by [1]

˙vn=Tn

bfb+gn−([ωn

en×] + 2 [ωn

ie×]) vn,(39)

where Tn

b∈R3×3is the transformation matrix from body

frame to the navigation frame. fb∈R3×1is the accelerometers

vector expressed in the body frame, gn∈R3×1is the gravity

vector expressed in the navigation frame. ωn

en is the angular

velocity vector between the earth centered earth ﬁxed (ECEF)

frame and the navigation frame. The angular velocity vector

between ECEF and the inertial frame is given by ωn

ie and the

rate of change of the transformation matrix is given by [1]

˙

Tn

b=Tn

bωb

ib×−ωb

in×,(40)

where ωb

ib =p q r T∈R3×1is the angular velocity

vector as obtained by the gyroscope and ωb

in is the angular

velocity vector between the navigation frame and the inertial

frame expressed in the body frame. The angular velocity

between the navigation frame and the inertial frame expressed

in the navigation frame is given by ωn

in. The alignment

between body frame and navigation frame can be obtained

from Tn

b, as follows

ε=

ϕ

θ

ψ

=

atan2Tb

n31,Tb

n32

arccos Tb

n33

−atan2Tb

n13,Tb

n23

∈R3×1,(41)

where ϕis the roll angle, θis the pitch angle, and ψis the

yaw angle. These three angles are called Euler angles. The

system matrix, F, is given by

F=

Fvv Fvε Tn

b03×3

Fεv Fεε 03×3Tn

b

03×303×303×303×3

03×303×303×303×3

(42)

where Tn

bis calculated by (18), and Fij ∈R3×3can be found

explicitly in the classical literature (see [1], [20], [34]).

The dynamic matrix terms Fij are provided:

Fεv =

0−1

RN+ˆ

h0

1

RM+ˆ

h0 0

0tan(ˆ

φ)

RN+ˆ

h0

(43)

Fvε =

0fD−fE

−fD0fN

fE−fN0

(44)

where matrix terms are the speciﬁc forces in navigation frame.

Fεε =

0ωD−ωE

−ωD0ωN

ωE−ωN0

(45)

where,

ωN

ωE

ωD

=

˙

ˆ

λ+ωiecos ˆ

φ

−˙

ˆ

φ

−˙

ˆ

λ+ωiesin ˆ

φ

(46)

The Fvv matrix columns are given as follows

F(1)

vv =

ˆvD

Re

−ωD−ωie sin ˆ

φ

2ˆvN

Re

F(2)

vv =

2ωD

ˆvD

Re+ˆvN

Retan ˆ

φ

−2ωN

F(3)

vv =

−ˆvN

Re

ωN+ωie cos ˆ

φ

0

(47)

ADAPTIVE STEP SIZE LEARNING WITH APPLICATIONS TO VELOCITY AIDED INERTIAL NAVIGATION SYSTEM / OR AND KLEIN 11

The shaping matrix is given explicitly by

G=

Tn

b03×303×303×3

03×3Tn

b03×303×3

03×303×3I3×303×3

03×303×303×3I3×3

(48)

B. Evaluation Criterions

1) AuC: Binary decision problems are commonly evaluated

using the receiver operating characteristic (ROC) curve. One of

the large advantages of the ROC is its representation capability

of accuracy of the test data. The ROC is a plot of sensitivity

vs. speciﬁcity. These two parameters are also known as:

TPR =T P

T P +F N (49)

and,

FPR =F P

F P +T N (50)

where TPR is the true positive rate (sensitivity), FPR is the

false positive rate (speciﬁcity). Pis the amount of positive

values, Nis the amount of negative values, T P is the number

of true positive, T N is the number of true negative, FP

is the number of false positive (type one error), and F N

is the number of false negative (type two error). The AuC,

Area under the Curve, is a measure of the two-dimensional

area underneath the entire ROC curve. This measure is scale-

invariant and classiﬁcation threshold invariant. Hence, it is a

very useful criterion for classiﬁcation performance evaluation.

2) Accuracy: The second criterion used in order to evaluate

the proposed models is the accuracy measure, given by:

ACC =T P +TN

T P +T N +F P +F N (51)

The accuracy is used as a measure of ”how well a binary

classiﬁcation test correctly identiﬁes a condition”. It compares

estimates of pre and post test probability. This is the ratio of

the number of true classiﬁed examples over the total number

of examples.

C. Bi vs. Multi Classiﬁcation and Regression Formalization

The major beneﬁt of deﬁning the problem as a bi-

classiﬁcation predictor, is that we minimize the number of

“chattering” between many step size values (might lead to

unstable ﬁlter). Also, using two values, one big 0.04[s] and

the other small 0.002[s], presents clearly the computational

effort reduction. A confusion matrix of 10 step size classes

are presented here, to demonstrate the lower robustness of

considering too much classes for this task. This is part of

our initial analysis and is not included in the paper.

Also, we formulated this problem as a regression task, where,

unfortunately, the obtained trained models result in high

RMSE respectively to the step size (linear regression RMSE:

0.025, Tree RMSE: 0.016).

Fig. 13. Quadratic SVM classiﬁer for 10 classes of ∆t.

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Barak Or (Member, IEEE) received a B.Sc. degree

in aerospace engineering from the Technion–Israel

Institute of Technology, Haifa, Israel, a B.A. degree

(cum laude) in economics and management, and

an M.Sc. degree in aerospace engineering from the

Technion–Israel Institute of Technology in 2016 and

2018. He is currently pursuing a Ph.D. degree with

the University of Haifa, Haifa.

His research interests include navigation, deep learn-

ing, sensor fusion, and estimation theory.

Itzik Klein (Senior Member IEEE) received B.Sc.

and M.Sc. degrees in Aerospace Engineering from

the Technion-Israel Institute of Technology, Haifa,

Israel in 2004 and 2007, and a Ph.D. degree in Geo-

information Engineering from the Technion-Israel

Institute of Technology, in 2011. He is currently

an Assistant Professor, heading the Autonomous

Navigation and Sensor Fusion Lab, at the Hatter

Department of Marine Technologies, University of

Haifa. His research interests include data driven

based navigation, novel inertial navigation architec-

tures, autonomous underwater vehicles, sensor fusion, and estimation theory.