SLAM of View-based Maps using SGD

David Valiente, Arturo Gil, Francisco Amor

os and

Oscar Reinoso

System Engineering Department, Miguel Hern

andez University, 03202, Elche, Spain

Keywords:

Visual SLAM, Omnidirectional Images, SGD.

Abstract:

This work presents a solution for the problem of Simultaneous Localization and Mapping (SLAM) based on a

Stochastic Gradient Descent (SGD) technique and using omnidirectional images. In the ﬁeld of applications

of mobile robotics, SGD has never been tested with visual information obtained from the environment. This

paper suggests the introduction of a SGD algorithm into a SLAM scheme which exploits the beneﬁts of om-

nidirectional images provided by a single camera. Several improvements have been introduced to the vanilla

SGD in order to adapt it to the case of omnidirectional observations. This new SGD approach reduces the

undesired harmful effects provoked by non-linearities which compromise the convergence of the traditional

ﬁlter estimates. Particularly, we rely on an efﬁcient map representation, conformed by a reduced set of image

views. The ﬁrst contribution is the adaption of the basic SGD algorithm to work with omnidirectional observa-

tions, whose nature is angular, and thus it lacks of scale. Former SGD approaches only process one constraint

independently at each iteration step. Instead, we think of a strategy which employes several constraints simul-

taneously as system inputs, with the purpose of improving the convergence speed when estimating a SLAM

solution. In this context, we present different sets of experiments which have been carried out seeking for

validation of our new approach based on SGD with omnidirectional observations. In addition, we compare

our approach with a basic SGD in order to prove the expected beneﬁts in terms of efﬁciency.

1 INTRODUCTION

The problem of SLAM is an essential aspect in the

ﬁeld of mobile robots, since a representation of the

environment is necessary for navigation purposes.

The aim of building a map entails a complex process,

since the robot needs to build the map in an incre-

mental manner, while, simultaneously, calculating its

localization inside it. Achieving a coherent map is

problematic due to the noisy sensor data, which af-

fects the simultaneous estimation of the map and the

path followed by the robot.

Traditionally, SLAM approaches have been differ-

entiated according to the representation of the map,

the basic algorithm to compute a solution and the kind

of sensor to gather information of the environment.

Some examples reveal the extensive use of laser range

sensors to construct maps. In this context, two mod-

els were mostly used to estimate map representations:

2D occupancy grid maps based on raw laser (Stach-

niss et al., 2004), and 2D landmark-based maps fo-

cused on the extraction of features, described from

laser measurements (Montemerlo et al., 2002).

The use of visual information has lately set a new

tendency towards the utilization of cameras. These

sensors outperform former sensors such as laser in

terms of the amount of valid information to build the

map of the environment. Despite the fact that vision

sensors are not as precise as laser sensors, they pro-

vide a huge variety of information of the scene, as

well as they are less expensive, lighter and more ef-

ﬁcient in terms of consumption. In the latter group

there exists several alternatives according to the num-

ber of cameras introduced and their conﬁguration. For

instance, stereo-based approaches (Gil et al., 2010),

in which two calibrated cameras extract relative mea-

surements of a set of 3D visual landmarks, determined

by a visual description. Other approaches present

their estimation of 3D visual landmarks by using a

single camera. In (Civera et al., 2008; Joly and

Rives, 2010) an inverse depth parametrization is car-

ried out to initialize the coordinates of each 3D land-

mark since the distance to the visual landmark can-

not be directly extracted with a single camera. Some

other approaches (Jae-Hean and Myung Jin, 2003)

have also combined two omnidirectional cameras to

exploit the beneﬁts of a wider ﬁeld, as in the case of a

stereo-pair sensor.

Besides the sensor and the representation of the

map, the kind of algorithm is the main basis to ob-

329

Valiente D., Gil Aparicio A., Amorós Espí F. and Reinoso O..

SLAM of View-based Maps using SGD.

DOI: 10.5220/0004482603290336

In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 329-336

ISBN: 978-989-8565-71-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

tain a solution for the problem of SLAM. In this

sense, different SLAM algorithms have been ex-

tensively used, differentiating between online meth-

ods such as, EKF (Davison and Murray, 2002),

Rao-Blackwellized particle ﬁlters (Montemerlo et al.,

2002) and ofﬂine algorithms, such as, for example,

Stochastic Gradient Descent (Grisetti et al., 2007).

Thus, the success of an efﬁcient SLAM solution is

closely dependent on the suitability of the combi-

nation between data sensor information, map repre-

sentation model and the SLAM algorithm. A large

amount of research has been conducted to obtain the

map representation, dealing simultaneously with the

estimation of the position of a set of 3D visual land-

marks expressed in a general reference system (Davi-

son and Murray, 2002; Gil et al., 2010; Andrew

J. Davison et al., 2004; Civera et al., 2008). They

rely on the capability of an EKF ﬁlter to converge to

a proper solution for the problem of SLAM. More re-

cently (Valiente et al., 2012), also based on EKF, pro-

posed a different representation of the environment,

where the map is formed by the position and orienta-

tion of a reduced set of image views in the environ-

ment. The most distinctive novelty of this representa-

tion is the capability to compute a relative movement

between views, which allows to retrieve the local-

ization of the robot and simultaneously it provides a

compact representation of the environment by means

of a reduced set of views. Such technique establishes

an estimation of a state vector which includes the

map and the current localization of the robot at each

timestep k. The estimation of the transition between

states at k and k + 1 takes into account the wheel’s

odometry as initial estimate, and the observation mea-

surements gathered by sensors. The major problem-

atic experienced by these methods based on an EKF

is the difﬁculty to keep the convergence of the estima-

tion when gaussian errors appear in the observation,

since they usually cause important data association

problems (Neira and Tard

os, 2001). A visual obser-

vation as omnidirectional is susceptible to introduce

non-linearties and therefore responsible for those er-

rors.

Some approaches have demonstrated considerable

improvements thanks to the assumption of a novel

representation of the map (Valiente et al., 2012), com-

posed by a subset of omnidirectional images, denoted

as views. Each view is representative of a large area

of the map since it may encapsulate a huge amount of

visual information in its neighborhood. Considering

a reduced set of images, it leads to a simpliﬁcation

of the map representation. The scheme allows for a

simple-computation of an observation measurement:

given two images acquired at two poses of the robot,

a speciﬁc subset of signiﬁcant points between images

can be processed in order to get a motion transforma-

tion which establishes the movement between the two

poses of the robot. This approach mitigates the main

EKF’s efﬁciency issue related to the high number of

variables needed to estimate a solution at each itera-

tion. Here, the number of landmarks, seen as views, is

drastically reduced and thus the time consumption for

the state’s re-estimation too. On the other hand, the

problematic of convergence associated to EKF’s al-

gorithms still exists. In this work we rely on the SGD

algorithm, which it is known to outperform EKF’s

SLAM results. Our main goal is to beneﬁt from

our visual observation model adapted to this kind of

SLAM solution algorithm based on SGD. Note that

even with SGD, solution’s convergence is not trivial

when the observation does not provide range distance

values. We also propose a variation in the procedure

to estimate the solution. Traditionally, either odom-

etry or observation are independently used in order

to iterate up to reach a ﬁnal solution. By contrast,

we suggest the use of the information provided by

several observations at the same time to quickly ﬁnd

the SLAM solution. This proposal gives the ﬁrst im-

pression of being liable to cause an increase of the

required computational resources. Nevertheless, we

have concentrated on the updating of several stages

of the iterative optimization process aiming at preven-

tion from possible harmful bottleneck effects.

The paper is divided as follows: Section 2 ex-

plains the SLAM problem in the mentioned context.

Then, in Section 3 we detail the speciﬁcations of a

Maximum Likelihood algorithm such as SGD. Sec-

tion 4 describes the main novelties provided by this

approach. Next, Section 5 shows the experimental re-

sults to test the suitability of the model and its bene-

ﬁts. Finally, Section 6 is referred to the result’s anal-

ysis in order to extract a general conclusion.

2 SLAM

A visual SLAM technique is expected to retrieve a

feasible estimation of the position of the robot in-

side a certain environment, which it has to be well-

determined by the estimation as well. In our ap-

proach, the map is composed by a set of omnidirec-

tional images obtained from different poses in the en-

vironment, denoted as views. These views do not rep-

resent any physical landmarks, since they will be con-

stituted by an omnidirectional image captured at the

pose x

= (x

,θ

) and a set of interest points ex-

tracted from that image. Such arrangement, allows us

to exploit the capability of an omnidirectional image

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330

to gather a large amount of information in a simple

image, due to its large ﬁeld of view. Thus, an im-

portant reduction is achieved in terms of number of

variables to estimate the solution.

The pose of the mobile robot at time t will be de-

noted as x

= (x

,θ

)

. Each view i ∈ [1, . . . , N]

is constituted by its pose x

= (x

,θ

)

, its uncer-

tainty P

and a set of M interest points p

expressed

in image coordinates. Each point is associated with a

visual descriptor d

, j = 1, . . . , M.

Thus, the augmented state vector is deﬁned as:

¯x =



··· x



(1)

where x

= (x

,θ

)

is the pose of the moving ve-

hicle and x

= (x

,θ

) is the pose of the N-view

that exist in the map.

2.1 Observation Model

According to the view-based representation of the

map, it is also necessary to describe an observation

model. The versatility of using omnidireccional im-

ages makes it possible to extract an observation mea-

surement which deﬁnes the motion transformation be-

tween two poses (Valiente et al., 2012), as seen in Fig-

ure 1. Actually, these poses represent the positions

where the robot acquired two speciﬁc images. To that

effect, only two images with a set of corresponding

points between them are required to obtain the trans-

formation. Following, the observation measurement

is described:





arctan(

−y

−x

) − θ

− θ

(2)

where the angle φ is the bearing at which the view

N is observed and β is the relative orientation be-

tween the images. The view N is represented by

= (x

,θ

), whereas the pose of the robot

is described as x

= (x

,θ

). Both measurements

(φ,β) are represented in Figure 1.

3 MAXIMUM LIKELIHOOD

MAPPING

3.1 Problem Speciﬁcations

A graph-oriented map is composed by a set of nodes

deﬁning the poses traversed by the robot and the land-

marks initialized into the map. The state vector s

encodes this representation through a set of variables

which are expressed in the following manner:



,θ

),(x

,θ

)... (x

,θ

)



(3)

Figure 1: Motion relation between two poses of the robot

with two omnidirectional images acquired at that poses.

The same motion relation is shown in the images with spe-

ciﬁc observation variables φ and β indicated.

being (x

,θ

) the 3D coordinates in a general ref-

erence system. A complementary subset of edges rep-

resents the relationships between nodes, by means of

either distance measurements generated by the odom-

etry or observations measurements provided by the

on board sensors. Both measurements are commonly

known as constraints and denoted as δ

, where j in-

dicates the observed node, seen from node i. The gen-

eral objective stated by these kind of methods (Olson

et al., 2006; Grisetti et al., 2007) is to minimize the

error likelihood expressed as:

(s) ∝ ηexp (−

( f

(s) − δ

)

Ω

( f

(s) − δ

))

(4)

being f

(s) a function dependent on the state s

and

both nodes j and i. The difference between f

(s) and

expresses the error deviation between nodes. Such

error term is weighted by the information matrix:

Ω

= Σ

−1

(5)

where Σ

−1

is the associated covariance, aimed at in-

troducing the uncertainty of the measurements. The

assumption of logarithmic notation leads to:

(s) ∝ ( f

(s) − δ

)

Ω

( f

(s) − δ

) (6)

= e

(s)

Ω

(s) = r

(s)

Ω

(s) (7)

SLAMofView-basedMapsusingSGD

331

being e

(s) the error determined by f

(s)-δ

(s),

also denoted as r

(s) to emphasize its condition of

residue. Finally, the global problem seeks the min-

imization of the objective function which represents

the accumulated error:

F(s) =

∑

h j,ii∈G

(s) =

∑

h j,ii∈G

(s)

Ω

(s) (8)

where G = {h j

i,h j

i...} deﬁnes the subset of

particular constraint conforming the map, either per-

taining to odometry or observation measurements.

3.2 Problem Solution. SGD

Once the formulation of the problem has been pre-

sented, it has to be detailed the Stochastich Gradi-

ent Descent algorithm. The basic goal is to compute

in an iterative manner a closer estimation to reach

a valid solution for the SLAM problem. The basis

of a SGD method lays on the minimization of (8)

through derivative optimization techniques. The es-

timated state vector is obtained as:

t+1

= s

+ ∆s (9)

where ∆s expresses a certain update with respect to s

term which is sequentially generated by means of the

constraint optimization procedure. It is worth noting

that in a general case, this update is calculated inde-

pendently at each step by using only a simple con-

straint, that is to say ∆s

= f (δ

). The general ex-

pression for the transition between s

and s

t+1

has the

following form:

t+1

= s

+ λ · H

−1

Ω

(10)

• λ is a learning factor to re-scale the term

−1

Ω

. Normally, λ takes decreasing val-

ues following the criteria λ = 1/n, where n is the

iteration step. This strategy pretends to quickly

reach the ﬁnal solution for higher values of λ. In

case it is close to the optimum, lower values of λ

are expected to prevent from oscillations around

the ﬁnal solution.

• H is the Hessian matrix, calculated as J

ΩJ,

and it represents the shape of the error function

through a preconditioning matrix to scale the vari-

ations of J

. According to (Grisetti et al., 2009),

H can be computed:

H ≈

∑

hi, ji

Ω

(11)

approximating H

−1

to allow a simpler and faster

computation:

−1

≈ [diag(H)]

−1

(12)

• J

is the Jacobian of f

with respect to s

, J

∂ f

∂s

. It converts the error deviation into a spacial

variation.

• Ω

is the information matrix associated to a con-

straint. Ω

= Σ

−1

, being Σ

the covariance matrix

corresponding with the constraint observations δ

This scheme updates the estimation by computing

the rectiﬁcation introduced by each constraint at each

iteration step respectively. Despite the learning factor

to reduce the weight by which each constraint updates

the estimation, the procedure can lead to an inefﬁcient

method to reach a stable solution, since undesired os-

cillations may occur due to the stochastic nature of

the constraints’ selection. For this reason, we pro-

pose an optimization process which takes into account

several constraints at the same iteration. It might be

thought that same drawbacks of a general case could

arise, in addition of some other inconveniences such

as undesired overloads of time, as a consequence of

the simultaneous processing of several constraints at

the same iteration. However, we have modiﬁed some

calculations at speciﬁc stages of the algorithm in or-

der to maintain the time requirements and even to re-

duce them. As a results, we achieved improved con-

vergence ratios in terms of speed. Further details will

be provided in the next sections.

4 MODIFIED SGD

The ﬁrst assumption to consider is the redeﬁned state

vector s

, which will be treated as a set of incremental

variables. Pose incremental is deﬁned as:

inc







dθ

...

dθ













− x

− y

− θ

− x

− y

− θ

...

− x

n−1

− y

n−1

− θ

n−1







(13)

where (dx

,dy

,dθ

) encode the variation between

consecutive poses in coordinates of the global refer-

ence system. A global encoding has the main draw-

back of not being capable to update more than one

node and its adjacents per constraint. Regarding a rel-

ative codiﬁcation of the sate, it arises the problem of

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332

non-linearities in J

. By contrast, an incremental state

vector allows a single constraint to generate a varia-

tion on every pose. In this framework, ∆s (9) affects

all poses because the state vector is differentially en-

coded.

It should be noticed that in this approach we are

dealing with a visual observation given by an om-

nidirectional camera. This fact makes us to adapt

the equations deﬁned in the previous section, since

the nature of the constraints are not only metrical

like odometry’s constraints. According to (2), given

two nodes, the observation measurement allows us to

determine a speciﬁc motion transformation between

them up to a scale factor. Therefore, the omnidirec-

tional measurements and the incremental represen-

tation require the reformulation of several terms in-

volved in the estimation. Following, we detail all the

terms which needed to be modiﬁed and recalculated.

• The ﬁrst adaption made was in f

, differentiating

between odometry:

odo

j,i

(s) =





dθ









j−1

dθ

j−1





+ ··· +





dθ





(14)

where (dx

,dy

,dθ

) are deﬁned in (13), and vi-

sual observation constraints:

visual

j,i

(s) =





arctan(

−dy

−dx

) − dθ

dθ

− dθ

(15)

where φ and β are directly computed as the obser-

vation measurement (Valiente et al., 2012) which

expresses the relation between the omnidirec-

tional images and the pose codiﬁcation (13). In-

spection of Figure 1 leads to deﬁne (2) and simi-

larly (15).

• Second step, according to this reformulation, is to

recalculate J

∂ f

∂s

. Please note the importance

of considering the indexes of the corresponding

nodes, either j > i or j < i since the derivatives

vary its form considerably. Furthermore, as seen

above, the dimensions of f

j,i

are different, fact

which has to be taken into account to resize the

rest of the terms involved in the SGD algorithm.

j,i

∂ f

j,i

∂s

∂ f

j,i

(s)

∂s

∂ f

j,i

(φ)

∂s

∂ f

j,i

(β)

∂s

(16)

• Thirdly, in this work we suggest the estimation

of the new state s

t+1

by considering several con-

straints at the same time. With this assumption,

we seek for more relevance of constraints’ weight

when searching for the optimal minimum estima-

tion. It is clear that computing more than one

constraint at each algorithm step leads to a cer-

tain overload. By contrast, with this approach, we

reduce the expensive estimation of H. In a gen-

eral case H is computed whenever a single con-

straint is introduced, that is to say, H is computed

as many times as constraints exist. In our case

we only compute H once for each subset of con-

straints introduced simultaneously into the sys-

tem. Consequently we obtain H in a more efﬁ-

cient manner, and therefore we compensate possi-

ble time overloads associated.

5 RESULTS

We have carried out three different experimental sets.

First, in Section 5.1 we show SLAM results obtained

in a simulated scenario to conﬁrm the validity of the

new SLAM approach supported by SGD. Then, Sec-

tion 5.2 presents a comparison of SLAM results us-

ing our approach and a basic SGD algorithm, respec-

tively. Finally, in Section 5.3 we present SLAM re-

sults in an ofﬁce-like environment.

5.1 Experimental Set 1

Assuring the convergence of an SLAM algorithm is of

paramount importance when a new solver technique

is introduced, such SGD. Moreover, the performance

of the new method has to consider certain modiﬁca-

tions to treat with a visual observation model, which

usually adds non-linearities. Figure 2(a) presents a

random simulation environment of 20x20m, where

the robot traverses 300m approximately. The real

path followed by the robot is shown with continuous

line, the odometry is represented with dash-dot line,

whereas the estimated solution is shown with dash

line. A set of views have been placed randomly along

the trajectory. The arrangement of these views is con-

trolled by an appearance ratio between images, so as

to assure a realistic placement of each view. A grid of

circles represents the possible poses where the robot

might move to and gather a new view. The number of

iterations of the SGD algorithm is 25. As it can be ob-

served in Figure 2(a), starting from a noisy ﬁrst odom-

etry estimate, the ﬁnal estimation has been rectiﬁed

following the tendency of the real path. Figure 2(b)

shows the decreasing evolution of the accumulated er-

ror probability P

(s) in (4), expressed in logarithmic

terms (8), versus the number of iterations. It can be

conﬁrmed the reliability of this new approach to work

SLAMofView-basedMapsusingSGD

333

−10 −5 0 5 10 15 20

−5

X(m)

Y(m)

Solution

Odometry

Ground truth

(a)

0 5 10 15 20 25 30

2500

5000

Iteration Step

F(s)

(b)

Figure 2: Figure 2(a) presents a map obtained with the proposed approach in an environment of 20x20m. Continuous line

shows the ground truth, dash-dot line the odometry and dash line the estimated solution. Figure 2(b) shows the accumulated

error probability F(s) along the number of iterations.

with omnidirecctional observations since it provides

a correct solution.

5.2 Comparing Experiments

The following experiments has been conducted to

compare our approach with a basic SGD in terms of

efﬁciency. We suggest a strategy to introduce simul-

taneously several constraints into the SGD algorithm.

The main goal is to improve the speed by which the

method iteratively optimizes up to reach a ﬁnal solu-

tion. In this sense, we have performed a SLAM exper-

iment, where the robot traverses 50m through a given

environment. Again, the number of views in the map

has been randomly placed by following the same pol-

icy explained above. The same experiment has been

repeated 200 times by using the same series of odom-

etry inputs, so as to provide mean values which ex-

press consistent results. Both approaches, ours and

the basic SGD algorithm’s have been compared. We

have modiﬁed the number of views N that the robot

is able to observe from each pose. The observation

range r of the robot has also been varied. Figure 3

presents results for the accumulated error probability

F(s), being the objective function which the SGD al-

gorithm seeks to minimize. We compare the solution

obtained with our approach, drawn with continuous

line, versus the solution obtained with a basic SGD

algorithm, drawn with dashed line. Figures 3(a), 3(b)

and 3(c) represent F(s) when the robot observes N=2,

N=4 and N=8 views, respectively. Since we look for a

fair comparison, the x-axis, originally representing it-

eration steps, has been transformed to a normalized

time variable to generate a trustworthy comparison

between both schemes. In terms of efﬁciency, it may

be proved that the solution provided by our method

outperforms the solution given by a basic SGD, since

the decreasing slope of F(s) is clearly steeper, there-

fore a quicker convergence speed is demonstrated.

This is the main consequence of combining several

constraints at each iteration step, instead of using only

one like a basic SGD used to. It is also remarkable

the relevance of the observation range of the vehicle

r. As seen in Figures 3(a), 3(b) and 3(c), long val-

ues of r provide a better convergence to the detriment

of shorter r. Since the omnidirectional observation is

angular, and it lacks of scale, views seen by the robot

at long distances in the map permit to compute a more

feasible localization.

5.3 Experimental Set 2

The purpose of this experiment is to deal with a more

realistic situation, deﬁned through an ofﬁce-like en-

vironment. Typical obstructor elements like walls or

doors have been introduced. Figure 4 describes an en-

vironment of 20x40m which the robot moves through.

The continuous line represents the real path followed

by the robot, the dash-dot line shows the odometry,

whereas the estimated solution with our approach is

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334

0 1 2 3 4 5 6

1000

2000

3000

4000

Normalized time

F(s)

min

inter

max

min

inter

max

(a)

0 1 2 3 4 5 6

1000

2000

3000

4000

Normalized time

F(s)

min

inter

max

min

inter

max

(b)

0 2 4 6 8 10 12

1000

2000

3000

4000

Normalized time

F(s)

min

inter

max

min

inter

max

(c)

Figure 3: Accumulated error probability F(s) versus time

in a SLAM experiment. Continuous lines show results pro-

vided by the proposed solution whereas the dash lines show

results provided by a basic SGD solution. Figures 3(a), 3(b)

and 3(c) show the results when the number of views ob-

served by the robot is N = 2, N = 4 and N = 8 respectively.

Different length observation ranges r

min

, r

iter

and r

max

are

deﬁned.

−10 −5 0 5 10 15 20 25 30

X(m)

Y(m)

Ground truth Solution Odometry

Figure 4: SLAM results in an ofﬁce-like environment of

20x40m. Continuous line shows the ground truth, dash-dot

line the odometry and dash line the estimated solution.

0 2 4 6 8 10 12

Normalized time

F(s)

Proposed solution

Basic SGD solution

Figure 5: Comparison of the accumulated error probability

F(s). Continuous line and dash line show F(s) provided by

the proposed approach and a basic SGD algorithm, respec-

tively.

described by a dash line. It may be noticed that in

only 15 iterations of the algorithm the robot is able

to estimate a quiet reliable solution, whose topology

follows the real path’s. On the other hand, the error of

the odometry grows out of bounds. Figure 5 shows a

comparison of the evolution of the accumulated error

SLAMofView-basedMapsusingSGD

335

F(s) versus the time. Once again, it is revealed the

improved capability of our approach to quickly reach

a solution, thus involving a better efﬁciency. In this

particular case, it can be observed that our approach

requires approximately less than 6 times the compu-

tational needs of a basic SGD to reach an optimum

value.

6 CONCLUSIONS

This work has presented an approach to the SLAM

problem by introducing a SGD algorithm adapted to

work with visual observations. The adoption of SGD

has been intended to overcome instabilities and harm-

ful effects which compromise the convergence of tra-

ditional SLAM algorithms such as ﬁlters like EKF.

These problems are mainly caused by a visual obser-

vation, whose nature is non-linear, especially distinc-

tive in omnidirectional observations. We rely on a vi-

sual SLAM approach which builds a map by using

a reduced set of omnidirectional image views. The

retrieval of the pose of the robot is provided by a

single-computation of two views, which directly pro-

vides a motion transformation to relate two poses of

the robot. A basic SGD algorithm has been modi-

ﬁed to work with such unscaled observation model.

In addition, a new strategy has been deﬁned to uti-

lize several constraints simultaneously into the same

SGD iteration, instead of utilizing only one per itera-

tion as former SGD does. We have shown SLAM re-

sults that prove the validity of this new approach with

omnidirectional observations. Moreover, we have es-

tablished a comparison between the results obtained

with our approach and those obtained with a basic

SGD algorithm. Consequently, the suitability of the

approach and its performance has been demonstrated,

but also its beneﬁts in the sense of efﬁciency.

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