A Loosely-Coupled Approach for Metric Scale Estimation in Monocular
Vision-Inertial Systems
Ariane Spaenlehauer
Vincent Fr´emont
Y. Ahmet S¸ekercio˘glu
Isabelle Fantoni
Abstract— In monocular vision systems, lack of knowledge
about metric distances caused by the inherent scale ambiguity
can be a strong limitation for some applications. We offer
a method for fusing inertial measurements with monocular
odometry or tracking to estimate metric distances in inertial-
monocular systems and to increase the rate of pose estimates.
As we performed the fusion in a loosely-coupled manner, each
input block can be easily replaced with one’s preference, which
makes our method quite flexible. We experimented our method
using the ORB-SLAM algorithm for the monocular tracking
input and Euler forward integration to process the inertial
measurements. We chose sets of data recorded on UAVs to
design a suitable system for flying robots.
I. INTRODUCTION
In recent times, research interest for monocular vision
has been strongly increasing in robotics applications. The
use of vision-based sensors such as cameras have numer-
ous advantages. They have low energy consumption, they
can be manufactured in very small sizes and their cost is
dramatically reducing every year. Their typical applications
include autonomous navigation, surveillance or mapping. A
key issue that directly impacts on the success of these appli-
cations is the estimation of locations and distances by using
the information gathered by these visual sensors. Several
studies show that combining visual information with low-
cost, widely available inertial sensors, Inertial Measurement
Units (IMUs), improves the accuracy of these estimations.
In this paper, we focus on this kind of sensor sets, called
“inertial-monocular” systems, which are composed of a
monocular camera and an IMU attached to Unmanned Aerial
Vehicles (UAVs). We present a computationally lightweight,
and fast solution for estimating the metric distances over
the visual information collected by the monocular camera of
a UAV. The problem is summarized as follows: By using
the frames provided by the camera, algorithms for odometry
or Simultaneous Localization And Mapping (SLAM) [1]
can estimate the camera positions and orientations (camera
poses) and, for the SLAM, create a 3-D representation of
the environment. However, the estimates are calculated up
to scale [2]. This scale ambiguity is inherent to monocular
vision and cannot be avoided. When a 3-D scene is captured
by the camera and projected into a 2-D frame, depth infor-
mation is lost. By measuring the same scene from different
points of views, depth can be reconstructed up to scale.
The scale factor is different for each frame, nevertheless
Sorbonne Universit´es, universit´e de technologie de Compi`egne, CNRS,
UMR 7253 Heudiasyc-CS 60 319, 60 203 Compi`egne Cedex, France
(e-mail: ariane.spaenlehauer@hds.utc.fr, vincent.fremont@hds.utc.fr, asek-
erci@ieee.org, isabelle.fantoni@hds.utc.fr).
recent algorithms provide consistent camera pose estimates,
which include the estimation of this scale factor. However,
the estimation of the scale factor does not provide metric
distances. The scale factor is used to ensure consistency in
the estimation of camera positions, i.e., large distances in the
world coordinate frame measured from a frame Fi remain
large even if they are measured again from another frame
Fj. To recover metric distances, the length of the camera
position vector has to be rescaled using a coefficient. This
scaling operation results in metric estimates for distances.
The aim of our method is to compute this scaling coefficient.
As mentioned above, monocular vision systems cannot re-
cover the scale of the world; therefore, at least one additional
sensor capable of measuring or estimating metric distances
must be added to the system. Several sensors can meet this
requirement such as lidar, ultrasound or IMU. The use of
IMU is often preferred in UAVs because of its small size
and low cost. However, IMU does not measure distances
directly but acceleration and angular velocities in the inertial
frame. Distances can be recovered through the calculation
of positions by integrating the acceleration measurements
but, consequently, the estimates drift quickly with the error
accumulation, which prevents any long-term integration.
The approach we propose is based on distances (L2 norm
of translation vectors) and is suitable to fuse the output of
any monocular odometry or the tracking part of SLAM algo-
rithms with inertial measurements. An overview of the sys-
tem architecture is shown in Fig. 1. In the following sections,
we first provide an overview of the leading approaches. Then,
mathematical details of the estimation of scaling coefficient
by using IMU measurements are presented. Finally, we test
the validity of our method over a set of UAV trajectories [3].
Fig. 1.
Overview of the system architecture: The blocks in grey can
be replaced with one’s preferences. For our experiments, we used the
ORB-SLAM algorithm for monocular pose estimation and Euler forward
integration for inertial pose estimation.
II. RELATED WORKS
Two main approaches for monocular visual-inertial fusion
can be distinguished in the literature: Loosely-coupled filter-
arXiv:1707.07518v2  [cs.RO]  9 Oct 2017
ing [4] [5] [6] and tightly-coupled systems [7] [8].
In tightly-coupled approaches, the fusion is done at a
low level of the system. Therefore, this requires a deep
understanding of the involved algorithms and specific design
for the system.
The method described in [8] is the inertial extension of
the DPPTAM [9], a direct SLAM algorithm. The tracking
thread is modified to include the IMU measurements. The
Gauss-Newton optimization is used to minimize the intensity
and IMU residuals. The state vector is composed of the
position, orientation and velocity of the robot and the IMU
biases. The IMU measurements are integrated between two
consecutive keyframes. The IMU residuals are the error of
the inertial integration between two keyframe with regard
to the state value at the corresponding time. The intensity
residuals are the photometric error between two keyframes.
They are calculated by reprojecting the map points in the
keyframes using the estimate of the relative camera pose.
The optimization of both residuals provide the final pose
estimate of the current keyframe with regard to the world
coordinate frame.
The method described in [7] is the inertial extension
of ORB-SLAM [10]. In ORB-SLAM, no functionality is
provided to calculate the uncertainty of pose estimates.
Therefore, the implemented method needs to avoid the direct
use of the uncertainty of the camera pose. To represent
information about uncertainty, the authors use information
matrices computed either from the preintegration of the IMU
measurements or from the feature extraction. The reprojec-
tion error and inertial error are minimized using the Gauss-
Newton optimization. The reprojection error comes from the
reprojection of map points in the current keyframe while
the IMU error is derived from the preintegration equations
described in [11].
In contrast to tightly-coupled approaches, in loosely-
coupled approaches, the vision part is considered as a black
box, only the output of the box is used. In most loosely-
coupled algorithms such as [4] [5] [6], the filter, which fuses
the measurements, is derived from Kalman Filtering, e.g.,
Extended Kalman Filter or Multi-State Constraints Kalman
Filter. The state is, at least, composed of the position,
orientation, velocity and biases of the IMU. The differential
equations, which govern the system and the IMU mea-
surements, are used to predict the state. The incorporation
of monocular visual measurements is done through the
measurement model when the Kalman gain needs to be
computed (the visual measurements update the state when
the innovation is calculated).
In [4], the state vector additionally includes the calibration
states (the constant relative position and orientation between
the IMU coordinate frame and the camera coordinate frame)
and a failure detection system. When a failure is detected
(abrupt changes in the orientation estimates with regard to
the measurement rate), the related visual measurements are
automatically discarded to prevent the corruption of data.
In [5], the authors use trifocal tensor geometry which
considers epipolar constraints in triples of consecutive im-
ages instead of pairs of images. Therefore, in addition to the
usual IMU states, the state vector also contains the pose and
orientation of the two previous keyframes.
In [6], the fusion is done by using measurements from
three sensors: In addition to the visual and inertial sensors,
a sonar is included in the system to measure distances (the
altitude between the UAV and the ground). IMU is used to
detect whether the UAV is flying level or tilted. If it is level,
the sonar measurements are directly used to estimate the
altitude. Otherwise, IMU measurements help to rectify the
incorrect altitude information due to the tilting of the UAV.
The scale factor estimation is represented as an optimization
problem between the sonar and visual altitude measurements
which is solved using the Levenberg-Marquardt algorithm
[12].
In this paper, we propose an approach for fusing monoc-
ular and inertial measurements in a loosely-coupled manner
which is simple to implement, requires small computational
resources and so, is suitable for UAVs. We decided to design
a loosely-coupled approach to make the methods used for
visual tracking and IMU measurement integration easy to
replace with any other method ones may prefer, which
ensures better flexibility and usability for our approach. In
our studies, we used the ORB-SLAM algorithm [10] for the
visual tracking part and Euler forward integration for the
inertial measurements processing.
III. SCALING COEFFICIENT ESTIMATION WITH
IMU MEASUREMENTS
A. Coordinate Frames
Our system (see Fig. 1) is composed of two sensors (a
camera and an IMU) attached on a rigid flying body, the
UAV. We distinguish four coordinate frames: camera {C},
vision {V }, inertial {I} and world {W} coordinate frames1.
The IMU measures data in {I} attached to the body of the
UAV. The integration of IMU measurements results in the
estimation of the pose of the IMU in {W}. The monocular
pose estimation algorithm outputs the camera poses in {V },
which corresponds to the first {C} coordinate frame when
the tracking starts, i.e.
{V } =
∧{C}t=0
(1)
The matrix ITC, which represents the transformation be-
tween {C} and {I}, is constant, and can be computed off-
line using a calibration method [13]. In the EuRoC dataset
sequences that we used for our experiments, ITC is already
provided. We consider that {W} corresponds to {I} at the
moment tracking starts, when the {V } coordinate frame is
generated, so
{W} =
∧{I}t=0
(2)
In the following paragraphs, we consider that the monocular
pose estimation algorithm outputs measurements in the world
coordinate frame by applying the formula
Wp = ITC
Vp
(3)
1The symbols used in the following paragraphs are given in Table I.
TABLE I
MATHEMATICAL NOTATION
Notation
Description
{A}
The coordinate frame {A} referred as A in equations.
ARB
3-by-3 rotation matrix that rotates vectors from {B} to
{A}.
t
3-by-1 translation vector.
ti
Translation vector calculated from inertial measurements.
tm
Translation vector calculated from monocular vision mea-
surements.
tg
Translation vector calculated from ground truth measure-
ments.
W tFi,Fj
Translation vector between the coordinate frames Fi and
Fj written in the world coordinate frame {W}.
ATB
Transformation matrix that transforms {B} into {A}.
ATBp
Transformation matrix that transforms {B} at time p into
{A}, implies that ATB is changing along time with
respect to {A}.
Fi
Camera coordinate frame associated with the image
frame i.
λ
Scaling coefficient.
ba, bω
IMU biases for the accelerometers and gyroscopes.
g
The gravity vector.
a(p)
3-by-1 vector which contains the accelerometer measure-
ments at time p (one vector component per axis).
ω(p)
3-by-1 vector which contains the gyroscope measure-
ments at time p (one vector component per axis).
∆T
Time step in seconds.
Ap
3-by-1 vector of a 3-D measurement in {A}.
where p is a 3-D measurement by the monocular pose
estimation algorithm.
B. Integration of IMU Measurements
Fig. 2.
IMU measurements obtained between the two subsequent image
frames are used to calculate the translation vectors and rotation matrices.
We assume having a monocular pose estimation algorithm
which outputs consistent camera poses in the world coor-
dinate frame. As a monocular camera cannot be used to
calculate metric distances, we use the IMU measurements to
compute the scaling coefficient. IMU is a sensor composed of
three accelerometers and three gyroscopes which respectively
measure accelerations and angular velocities along each
of the three axis of the inertial frame. The pose of the
IMU can be estimated by integrating the accelerometer and
gyroscope measurements. However, the integration of the
IMU measurement noise and the IMU biases makes the pose
estimates to drift fast. Therefore, IMU measurements must
be integrated only over a short period for limiting the drift
and consequently to corrupt the estimates. We integrate IMU
measurements between two consecutive image frames. So,
if Fk and Fk+1 are two coordinate frames associated with
image frames k and k+1 (see Fig. 2), the integration results
in the estimation of the rotation matrix FkRFk+1, velocity
W vFk,Fk+1 and translation W tFk,Fk+1 vectors between the
two consecutive frames Fk and Fk+1 in the world coordinate
frame using IMU measurements.
We integrated the IMU measurements using Euler forward
integration following the description given in [8]
FiRFj =
k+N−1
Y
p=k
expSO(3)([ω(p) + bω(p)]∧∆T)
(4)
where ω is the vector of gyroscope measurements, bω the
gyroscope bias, k the time step of frame Fi, k + N the time
step of frame Fj, N −1 the number of IMU measurements
between the consecutive frames Fi and Fj and ∆T is the
time step size (in seconds)
W vFi,Fj =
k+N−1
X
p=k
[W RIp(a(p) + ba(p)) −g]∆T
(5)
where
W RIp is the rotation matrix between the world
coordinate frame and the IMU coordinate frame at time
p, a is the vector of accelerometer measurements, ba the
accelerometer bias and g the gravity vector
W tFi,Fj =N W vFi∆T+
1
2
k+N−1
X
p=k
h
(2 (k + N −1 −p) + 1)
wR
p (a(p) + ba(p)) −g

 i
∆T 2
(6)
The expSO(3) operator maps an vector of so(3) to a matrix
of SO(3). The wedge operator .∧convert a 3 × 1 vector
into an element of so(3), i.e., a skew-symmetric matrix of
size 3 × 3. The IMU biases, ba and bω, were modeled as a
random walk process
ba(k + 1) = ba(k) + ∆Tσ2
a
(7)
where σ2
a is the variance associated to the IMU accelerom-
eters
bω(k + 1) = bω(k) + ∆Tσ2
ω
(8)
where σ2
ω is the variance associated to the IMU accelerom-
eters
Note that the IMU integration equations (Eqs. 4, 5, 6) can
be replaced with another approach for numerical integration
(such as [11]) as long as this calculates the translation vector
between two consecutive frames in the world coordinate
frame {W}.
It is assumed that the estimates provided by the monocular
pose estimation algorithm xm drift slower than the estimates
xi computed using the IMU measurements, i.e., xm is more
accurate than xi. At each incoming frame, we used the value
of xm to initialize xi. We observed that a good initialization
for the IMU estimates can greatly improve the accuracy
of the estimation of the scaling coefficient. However, the
initialization of the estimates xi is not discussed in this paper
and is part of the further improvement we plan to do.
We also benefit from the high measurement rate of the
IMU (between 100 Hz and 200 Hz) to provide fast pose
estimates. The pose estimates from the vision algorithm xm
can be updated once per new frame at maximum. Therefore,
the camera pose is updated at the frame rate, usually around
30 Hz, which can be too slow for some applications such as
control or navigation.
C. Calculation of Scaling Coefficient
We want to find the scaling coefficient λ as follows
∥Wti∥2 = λ∥Wtm∥2
(9)
where Wti are the translation vectors of the camera position
given by the integration of IMU measurements in the world
coordinate frame, Wtm are the translation vectors of the
camera position given by the monocular odometry or SLAM
algorithm in the world coordinate frame and ∥.∥2 is the L2
norm of a vector.
For each incoming new frame Fj, the translation of
the camera between the consecutive frames Fi and Fj in
the world coordinate frame given by the monocular vision
algorithm Wtm
Fi,Fj is measured. We then integrate the cor-
responding IMU measurements using the Eq. (1), (2) and
(3) to obtain the corresponding translation from the inertial
measurements Wti
Fi,Fj
λFi,Fj =
∥Wti
Fi,Fj∥2
∥Wtm
Fi,Fj∥2
(10)
So Eq. 10 provides an estimated value of the scaling coeffi-
cient λ.
We can measure λ for each frame, but the measurement
noise on each measurement is significant because both the
outputs of the SLAM algorithm and the IMU integration
drift. Four methods have been tested to calculate the scal-
ing coefficient ˆλ using the measurements λFi,Fj. The four
methods are: moving average on λFi,Fj with an additive
model for the error, moving average on log(λFi,Fj) with
a multiplicative model for the error, an autoregressive Filter
and a Kalman Filter.
The moving averages are calculated over the available
measurements at time t
ˆλ1 = 1
M
M−1
X
k=2
 
∥Wti
Fk,Fk+1∥2
∥Wtm
Fk,Fk+1∥2
!
(11)
where the error model is additive
ˆλ2 = exp
 
1
M
M−1
X
k=2
log
 
∥Wti
Fk,Fk+1∥2
∥Wtm
Fk,Fk+1∥2
!!
(12)
where the error model is multiplicative and M is the number
of frames at the considered discrete time t. The first frame
is skipped because the error on the first IMU measurement
is generally large.
We also decided to implement an autoregressive filter
(AR) to estimate ˆλ. Moreover, this filter can be used to
check whether the measurements are correlated in time. The
current value of the filter output y(i) is a weighted linear
combination of the p previous outputs (p is the order of
the filter) and the current measurement. The weights are
computed by solving the Yule-Walker equations
y(i) = K + s(i) +
p
X
j=1
αiy(i −j)
(13)
where y(i) is the output of the AR filter at discrete time i, αi
are the weights calculated with the Yule-Walker equations,
s is a zero-mean random variable with
s(i) = λ(i) −K
(14)
λ(i) =
∥Wti
Fi,Fi+1∥2
∥Wtm
Fi,Fi+1∥2
(15)
The weights αi and bias term K are calculated solving


µ
c1
c2
...
cp


=


1
µ
µ
. . .
µ
µ
c0
c1
. . .
cp−1
µ
c1
c0
. . .
cp−2
...
...
...
...
...
µ
cp−1
cp−2
. . .
c0




K
α1
α2
...
αp


(16)
where cp is the cross-correlation of the signal y with temporal
lag p and µ is the average of y at time i.
The AR filter diverged and it led to poor accuracy for the
estimate ˆλ in all EuRoC sequences. These results show that
the measurements are not temporally correlated and do not
follow an autoregressive model.
Finally, a Kalman Filter has been implemented to estimate
ˆλ. The model is
λk = aλk−1 + wk
(17)
zk = hλk−1 + vk
(18)
where a = 1 and h = 1.
The prediction step is done using
ˆλk|k−1 = aˆλk−1|k−1
(19)
and the a priori variance pk|k−1
pk|k−1 = a2pk−1|k−1 + q
(20)
where q is the covariance of the model white noise w.
The correction step is calculated as follows
kk|k =
hpk|k−1
h2pk|k−1 + r
(21)
where k is the Kalman gain and r is the covariance of the
measurement white noise v
ˆλk|k = ˆλk|k−1 + kk|k(zk −hˆλk|k−1)
(22)
and the a posteriori variance pk|k
pk|k = pk|k−1(1 −hkk|k)
(23)
IV. EXPERIMENTAL RESULTS
We experimented the proposed method using the se-
quences from EuRoC dataset [3] and the ORB-SLAM al-
gorithm [10]. The EuRoC dataset provides eleven sequences
recorded by an Asctec Firefly hex-rotor helicopter in two
different environments, a room equipped with a Vicon mo-
tion capture system and a machine hall. We used the frames
from one of the front stereo camera (Aptina MT9V034
global shutter, WVGA monochrome, 20 FPS) to emulate
monocular vision and the measurements of the MEMS IMU
(ADIS16448, angular rate and acceleration, 200 Hz). The
ground truth is measured either by the Vicon motion capture
system in the sequences recorded in the Vicon room, or by
a Leica MS50 laser tracker and scanner in the machine hall
environment.
In the following, the sequences referred as V1 01, V1 02
and V1 03 were recorded in the Vicon room with configura-
tion of texture 1; the sequences referred as V2 01, V2 02 and
V2 03 were recorded in the Vicon room with configuration
of texture 2; the sequences referred as MH01, MH02, MH03,
MH04 and MH05 were recorded in the machine hall using
the Leica system. Note that the trajectory of the UAV is
different in each sequence.
The estimation of the scaling coefficient during the se-
quence V1 01 is pictured in Figure 3. The value of the
scaling coefficient can be compared to a ground truth value,
which is computed using a moving average with Vicon (or
Leica) measurements Wtg instead of IMU measurements
λg = 1
M
M−1
X
k=1
 
∥Wtg
Fk,Fk+1∥2
∥Wtm
Fk,Fk+1∥2
!
(24)
The error eλ between the ground truth scaling coefficient λg
and the coefficient we estimate using inertial measurement
ˆλ is calculated as follows
eλ = ∥λg −ˆλ∥1
(25)
where ∥.∥1 is the L1 norm.
We rescaled the trajectory provided by the monocular
algorithm. As presented in Fig. 4, the distances given by the
monocular algorithm are arbitrary but consistent, therefore
the UAV’s trajectory is scaled differently than the ground
truth. In Fig. 5, we rescaled the monocular trajectory of the
sequence V1 01 using the ground truth λg and estimated
ˆλ1 scaling coefficients. We computed the root-mean-square
deviation (RMSE) for each sequence as follows
RMSE =
v
u
u
t
PM
i=1

λgxT
i −ˆλxT
i
 
λgxi −ˆλxi

M
(26)
where M is the number of frames in the sequence and x is
the position of the camera in the world coordinate given by
the monocular algorithm (ORB-SLAM for our experiments).
The RMSE for each EuRoC sequence is given in Tab. II.
The initial trajectory of the UAVs in the sequence V1 01
provided by the ORB-SLAM algorithm is displayed in Fig.
 1
 2
 3
 4
 5
 6
 7
 8
 9
 10
 0
 500
 1000
 1500
 2000
 2500
frame number
Ground Truth Scaling Coefficient
Estimated Scaling Coefficient
Fig. 3.
The estimation of the scaling coefficient λ using the sequence
V1 01 from EuRoC dataset (In blue the ground truth calculated from the
Vicon estimates, in red the proposed estimation method).
-2.5
-2
-1.5
-1
-0.5
 0
 0.5
 1
 1.5
 2
 2.5-3
-2
-1
 0
 1
 2
 3
 4
 0.8
 1
 1.2
 1.4
 1.6
 1.8
 2
(m)
Ground Truth
ORB SLAM
(m)
(m)
(m)
Fig. 4.
The trajectory of the camera during the V1 01 sequence from
EuRoC dataset in the world coordinate frame (ORB-SLAM measurements
are in blue, the ground truth measured with a Vicon motion capture system
are in red).
-3
-2
-1
 0
 1
 2
 3-3
-2
-1
 0
 1
 2
 3
 4
 0.8
 1
 1.2
 1.4
 1.6
 1.8
 2
(m)
Ground Truth Scaling
Estimated Scaling
(m)
(m)
(m)
Fig. 5.
The rescaled trajectory of the camera provided by ORB-SLAM
in the V1 01 sequence from EuRoC dataset in the world coordinate frame
(the blue trajectory was rescaled using the value of λ, the red trajectory was
rescaled using the ground truth scaling coefficient).
TABLE II
SCALING COEFFICIENT AND EFFECT ON THE TRAJECTORY
EuRoC
ˆλ
λg
eλ
RMSE (m)
Total distance (m)
Sequence
ˆλ1
ˆλ2
KF
ˆλ1
ˆλ2
KF
ˆλ1
ˆλ2
KF
Ground truth
Best estimate
V1 01
2.49
1.89
12.42
2.31
0.19
0.42
10.11
0.22
0.51
12.25
59.26
63.91
V1 02
1.80
1.41
7.57
2.35
0.55
0.94
5.22
0.55
0.95
5.25
76.58
58.74
V1 03
2.55
1.92
6.73
3.54
1.00
1.63
3.19
0.46
0.75
1.47
77.53
55.78
V2 01
2.92
2.33
1.17
3.02
0.10
0.70
1.86
0.09
0.60
1.67
36.93
35.66
V2 02
4.32
1.87
82.15
3.56
0.77
1.684
78.80
0.47
1.03
47.99
83.92
101.93
V2 03
1.94
1.42
1.26
2.15
1.84
2.36
2.52
1.66
2.13
2.27
139.15
71.30
MH01
251.58
7.57
10.10
6.92
208.66
0.65
3.18
216.55
0.68
3.30
85.50
93.54
MH02
51.91
2.64
1.08
2.94
48.97
0.30
1.86
146.69
0.90
5.56
84.39
75.88
MH03
35.81
2.31
4.93
3.77
32.04
1.46
1.16
39.85
1.82
1.45
131.13
171.45
MH04
36.74
3.78
9.28
7.51
29.23
3.74
1.17
32.78
4.19
1.98
103.58
127.94
MH05
33.21
2.05
2.27
2.74
30.47
0.69
0.47
109.16
2.48
1.68
116.02
96.22
4. The same trajectory rescaled using the scaling coefficients
λg and ˆλ1 is displayed in Fig. 5.
As expected, the bigger the scaling coefficient error eλ, the
bigger the RMSE. The sequences recorded in the Vicon room
provide trajectories with lower RMSE than the sequences
recorded in the machine hall. The difference between the two
sets of sequences can be explained by the strong excitation
of the IMU for the calibration of the Leica laser that
incorporates a lot of noise in the measurements ti. In the Vi-
con sequences, the estimate ˆλ1 outperforms. The sequences
recorded in the machine hall are very challenging for the
inertial fusion because the UAV performed very fast trans-
lational movements during a few seconds for Leica ground
truth calibration purposes which result in large acceleration
measurements and partially corrupt the inertial estimates as
shown in Fig. 6. Therefore, in the machine hall sequences
where strong noise corrupts some IMU measurements, ˆλ2
is far better than ˆλ1, which never managed to completely
absorb the strong perturbations of the calibration. Interest-
ingly, Kalman Filter (KF) gives also satisfactory results for
Leica sequences. The results of Kalman Filter can be further
improved with a finer tuning of the process noise variance
q. For instance, with a smaller value for q, the RMSE of
sequences MH 03 and MH 04 drops to 0.17 m and 0.76
m respectively. As every single EuRoC sequence is quite
different from the others, finding a nice tuning value for
the Kalman Filter is not straightforward. We recommend to
tune the filter accordingly to the type of flight the UAV
performs (smoothness and aggressiveness of trajectories,
motion speed, angular velocities). If a Kalman Filter cannot
be implemented or tuned, ˆλ2 remains a acceptable estimate.
More broadly, keeping the IMU out from large perturbations
by using smooth trajectories provides more accurate esti-
mates. The estimates computed through the AR filter, which
are not presented because of the dramatically large value
of RMSE, show that there is no temporal correlation of the
error.
V. CONCLUDING REMARKS AND FUTURE WORK
We presented a fast and easy-to-implement method for
the calculation of the scaling coefficient by fusing inertial
 0
 200
 400
 600
 800
 1000
 1200
 1400
 1600
 0
 500
 1000
 1500
 2000
 2500
frame number
Scaling Ratio
Ground Truth Scaling Coefficient
Estimated Scaling Coefficient
Fig. 6.
An example of the corruption of estimates due to the fast
translational movements (shown in the rectangular area as marked) for the
Leica calibration in one of the sequences recorded in the machine hall
environment (MH05).
measurements with monocular pose estimation. Monocular
camera systems, due to their nature, can not provide the real-
world scale of the pose estimates. To overcome this problem,
we use the inertial measurements produced by an IMU to i)
estimate the scaling coefficient, which relates the monocular
camera pose estimation to the real-world scale, and ii) speed
up the pose estimation by exploiting the availability of the
inertial measurements in very high rates.
The method is highly modular, which makes each compo-
nent to be easily replaceable with one’s preferences without
impacting the overall operation of the system.
To improve the current method, we plan to further in-
vestigate the initialization of the IMU integration process,
particularly with the incorporation of the current estimate of
the scaling coefficient when appropriate.
We defined three approaches for calculating the scaling
coefficient with regard to the nature of the trajectory followed
by the UAV. We found that the Kalman Filter approach
gives accurate estimates when the tuning is done well,
which unfortunately can be hard to do for some applications.
Determination of the tuning value of the process noise is a
complex topic which will be part of our future research work
and experimentations.
ACKNOWLEDGMENT
This work was carried out in the framework of the Labex
MS2T and DIVINA challenge team, which were funded by
the French Government, through the program Investments
for the Future managed by the National Agency for Research
(Reference ANR-11-IDEX-0004-02).
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