Performance characterisation of footmounted ZUPTaided INSs and other related systems
ABSTRACT Footmounted zerovelocityupdate (ZUPT) aided inertial navigation system (INS) is a conceptually well known with publications in the area typically focusing on improved methods for filtering and addition of sensors and heuristics. Despite this, the performance characteristics, which would ultimately justify and give guidelines for such system modifications of ZUPTaided INSs and other related systems, are in some aspects poorly documented. Unfortunately, the systems are nonlinear, meaning that the performance is dependent on the system setup, parameter setting, and the true trajectory. This complicates the process of evaluating performance and partially explains the few publications with detailed performance characterisation results. Therefore in this article we suggest and motivate methodologies for evaluating performance of ZUPTaided INS and other related systems, we apply them to a suggested baseline setup of the system, and study some aspects of the performance characteristics.

Conference Paper: Evaluation of zerovelocity detectors for footmounted inertial navigation systems
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ABSTRACT: A study of the performance of four zerovelocity detectors for a footmounted inertial sensor based pedestrian navigation system is presented. The four detectors are the acceleration moving variance detector, the acceleration magnitude detector, the angular rate energy detector, and a novel generalized likelihood ratio test detector, refereed to as the SHOE. The performance of each detector is assessed by the accuracy of the position solution provided by the navigation system employing the detector to perform zerovelocity updates. The results show that for leveled ground forward gait at a speed of 5 km/h, the angular rate energy detector and the SHOE give the highest performance, with a position accuracy of 0.14% of the travelled distance. The results also indicate that during leveled ground forward gait, the gyroscope signals hold the most reliable information for zerovelocity detection.Indoor Positioning and Indoor Navigation (IPIN), 2010 International Conference on; 10/2010  [Show abstract] [Hide abstract]
ABSTRACT: The localization of an ambulatory individual, a.k.a. a pedestrian, is a quicklydeveloping domain with the potential to permeate into a variety of applications, as knowledge of an individual's location within an environment becomes evermore useful. In order to automate the localization task, positioning modules are prime candidates for inclusion in a system. Such modules are expected to reduce both the effort and time incurred during the localization process while improving the accuracy and organization of the exchanged data. Building on and combining recent developments in the fields of step detection using inertial measurement units and structure from motion using a camera rig, the work presented in this paper is an implementation of a pedestrian localization system targeted specifically at infrastructureless indoor localization. The inertial measurement unit and camera rigs are respectively attached to the mobile user's foot and waist, and the collected data is processed by the localization module to obtain a current position. The focus of this paper is the implementation and preliminary testing of this localization module's components.01/2011;  SourceAvailable from: elib.dlr.de[Show abstract] [Hide abstract]
ABSTRACT: The use of footmounted inertial measurement units (IMUs) has shown promising results in providing accurate human odometry as a component of accurate indoor pedestrian navigation. The specifications of these sensors, such as the sampling frequency have to meet requirements related to human motion. We investigate the lowest usable sampling frequency: To do so, we evaluate the frequency distribution of different human motion like crawling, jumping or walking in different scenarios such as escalators, lifts, on carpet or grass, and with different footwear. These measurements indicate that certain movement patterns, as for instance going downstairs, upstairs, running or jumping contain more high frequency components. When using only a low sampling rate this high frequency information is lost. Hence, it is important to identify the lowest usable sampling frequency and sample with it if possible. We have made a set of walks to illustrate the resulting odometries at different frequencies, after applying an Unscented Kalman Filter (UKF) using Zero Velocity Updates. The odometry error is highly dependent on the drift of the individual accelerometers and gyroscopes. In order to obtain better odometry it is necessary to perform a detailed analysis of the sensor noise processes. We resorted to computing the Allan variance for three different IMU chipsets of various quality specification. From this we have derived a bias model for the UKF and evaluated the benefit of applying this model to a set of real data from walk.10/2013;
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2010 INTERNATIONAL CONFERENCE ON INDOOR POSITIONING AND INDOOR NAVIGATION (IPIN), 1517 SEPTEMBER 2010, Z¨URICH, SWITZERLAND
Performance characterisation of footmounted
ZUPTaided INSs and other related systems
JohnOlof Nilsson⋄, Isaac Skog, and Peter H¨ andel
Signal Processing Lab, ACCESS Linneaus Centre, KTH Royal Institute of Technology
Stockholm, Sweden. Email:⋄jnil02@kth.se
Abstract—Footmounted zerovelocityupdate (ZUPT) aided
inertial navigation system (INS) is a conceptually well known with
publications in the area typically focusing on improved methods
for filtering and addition of sensors and heuristics. Despite this,
the performance characteristics, which would ultimately justify
and give guidelines for such system modifications of ZUPT
aided INSs and other related systems, are in some aspects
poorly documented. Unfortunately, the systems are nonlinear,
meaning that the performance is dependent on the system setup,
parameter setting, and the true trajectory. This complicates the
process of evaluating performance and partially explains the few
publications with detailed performance characterisation results.
Therefore in this article we suggest and motivate methodologies
for evaluating performance of ZUPTaided INS and other related
systems, we apply them to a suggested baseline setup of the sys
tem, and study some aspects of the performance characteristics.
I. INTRODUCTION
Footmounted ZUPTaided INS is conceptually a well
known technique with numerous related publications over the
last decade, e.g.[1]–[5] and references therein. However,
the system behaviour and performance characteristics are in
some aspects poorly documented and understood. The main
reasons for this are likely the difficulty of constructing sensible
simulations of the system, the nonlinear nature of the system,
the multiple system parameters affecting the behaviour of the
system, and the high dynamic of the impact of the foot during
bipedal locomotion exciting error modes of the inertial sensors
rarely encountered in other aided INS. However, the current
status and trend of the technology, the fundamental technique
being well known and publications focusing on new filtering
methods and addition of more sensors and heuristic, would
benefit from knowledge about the performance characteristics
of different setups of the system. The reasons for this are
simple: First, to obtain guidelines for the development of
a system setup the performance characteristics of the set
up need to be understood; Second, a modification of a set
up ought to be motivated by an increased performance. For
this a baseline performance is needed which is given by the
performance characteristics of a baseline setup. However, our
experience is that, even for a rather narrow span of external
conditions, a wide range of performance can be achieved with
different parameter settings and vice versa. Without detailed
knowledge of the performance characteristics this might result
in erroneous conclusions about the effect on the performance
of system modifications. Therefore, in this article we set out to
make a structured approach to performance characterisation of
ZUPTaided INSs and other related systems. The performance
characteristic of a ZUPTaided INS setup we define to be the
functional dependence of performance on parameters affecting
performance. The setup we define to be the set of discrete
factors affecting performance, that is essentially the discrete
design choices determining the structure of a ZUPTaided
INS implementation. The parameters are obtained from a
parameterisation of the remaining factors of continuous nature
that affect the performance. Hence, the goal is to evaluate the
functional dependence of performance on the parameters of a
given setup.
Unfortunately, the ZUPTaided INS is a nonlinear system.
Thus the performance is ultimately determined by the system
setup, system parameter settings, and the true trajectory.
Therefore, it is not clear how to evaluate performance nor is it
clear how to handle the parameterisation of the system. Even
the concept of performance is not well defined for the type
of system in question. To handel this we begin in Section II
by identifying the fundamental structures and components of
a ZUPTaided INS. This will partially set the framework for
factors affecting performance and hence also the performance
characterisation. In Section III, we proceed by defining per
formance, drawing up guidelines for the parameterisation, and
deriving methods for estimating the performance. Finally, in
Section IV the methods are applied to a suggested baseline
setup and some aspects of the performance characteristics are
studied.
The main results of this article are: 1. A mathematically
motivated methodology to evaluate performance of ZUPT
aided INSs and other related systems. 2. An illustration and
analysis of the performance characteristics of a suggested
baseline setup over a range of internal and external parameter
settings.
II. STRUCTURE OF ZUPTAIDED INS SETUPS
The setup has been defined to be the discrete design choices
determining the structure of a ZUPTaided INS implementa
tion. The exact design choices will differ from different setups
since certain design choices will give rise to others. However,
a general structure and fundamental components of the system
can be identified, giving a structure to the setup specifications.
Conceptually, the ZUPTaided INS consists of an inertial
measurement unit (IMU) under the influence of a trajectory
9781424458646/10$26.00 c ⃝ IEEE
Page 2
Sensor
model
IMU
Mechanisation
equations
Dynamic
model
Fusion filter
Estimates
Trajectory
Fig. 1.
INS. The IMU is the only sensor, the sensor model relates the IMU output to
the ideal output, the mechanisation equations relate the ideal IMU output to
evolution of the navigational states, and the dynamic model adds knowledge
of the trajectory. The information from the building blocks are fused with the
fusion filter giving system state estimates.
The block diagram illustrates the building blocks of the ZUPTaided
and a filter giving system state estimates. It is assumed that
the ZUPTs are derived from the inertial measurements such
that there are no other inputs to the system.
A. Filtering setup
The filter in turn consists of models fused together by some
sensor fusion filter. Then the fundamental models of the ZUPT
aided INS can be identified as: the sensor model of the IMU,
[˜ u푘,b푘+1] = 푔(u푘,b푘,n1,푘);
(1)
the mechanisation equations (kinematic model) of the INS,
x푘+1= 푓(x푘,u푘,푑푡푘,n2,푘);
(2)
and the dynamic model,
[0,c푘+1] = ℎ(x푘,u푘,c푘,n3,푘);
(3)
including distributions of the stochastic components, n1,푘,
n2,푘, and n3,푘, possibly dependent on x푘, u푘, b푘, and c푘.
˜ u푘 is the IMU output, u푘 is the ideal IMU output (the true
specific force and angular rates), b푘 is sensor states, x푘 is
the navigation states, 푑푡푘is the sampling period of the IMU,
and c푘 is dynamics states. Further, the zerovelocityupdate
(ZUPT) attribute of the system implies that the dynamic model
will have the structure
[0,c푘+1] =ℎ(x푘,u푘,c푘,n3,푘)
=[ℎ′(x푘,u푘,c푘,n′
− [v푘,0];
where v푘 is the velocity (subcomponent of x푘), ℎ′(⋅) is the
zerovelocity detector, ℎ′′(⋅) is the dynamic state model, and
n′
of the ZUPTaided INS are illustrated in Fig. 1.
Additionally, the sensor fusion of aided INS is typically im
plemented with a complementary filter structure often making
use of some sort of Kalman filter (KF) [6]. Even though not a
defining attribute of ZUPTaided INS, it is still so commonly
3,푘),ℎ′′(x푘,u푘,c푘,n′′
3,푘)]
(4)
3,푘and n′′
3,푘are subcomponents of n3,푘. The building blocks
Sensor
model
Dynamic
model
IMU
Mechanisation
equation
KF
Trajectory
State estimates Covariance est.
INS
Complementary filter
ZUPT
()−1
Fig. 2.
a ZUPTaided INS setup with a complementary filter structure implemented
with a KF. The complementary filter estimates and feed back the errors of
the filter of the INS.
The block diagram illustrates the components and their relations of
used that its implications are worse mentioning. The comple
mentary structure adds yet another model derived from the
fundamental models. The estimation of the complementary
filter structure is based on a model for the evolution of the
errors of the state estimates rather than on models for the
evolution of the states themselves,
[훿x푘+1,훿b푘+1] = 푓훿(훿x푘,훿b푘, ˜ u푘) + n4,푘,
(5)
where 훿x푘+1and 훿b푘+1are the errors in the state estimates
and n4,푘is the combined effect of n1,푘, n2,푘, and linearisation
errors and other approximation made in the derivation of (5).
The structure of the ZUPTaided INS, assuming a comple
mentary filter structure, is illustrated in Fig. 2. Note that the
setup contains the structure of the models and the fusion filter
(discrete choices) but not necessarily all numerical values of
constants in the models and the fusion filter.
B. Hardware setup
Essentially the only hardware that will affect performance
is the IMU. The physical IMU cannot normally be varied
continuously in any way and would therefor be considered
to be a part of the setup. Possibly also the boot in use could
be included in the setup specifications. The IMU selection
might influence the selection of the sensor model. However,
the most important consideration of the IMU selection is to
choose an IMU with sufficient dynamic range or the sensor
model of the IMU will not be invertible over the operational
range.
III. EVALUATING PERFORMANCE OF ZUPTAIDED INS
To evaluate the performance: First, performance need to
be defined and a performance metric chosen; Second, factors
apart from the setup affecting the performance need to be
identified and parameterised; Third, methods for estimating
the performance need to be found and applied.
Page 3
A. Performance metric
Generally one speaks about performance of navigation
systems in terms of accuracy, integrity, availability, and
continuity of service. Being selfcontained, availability and
continuity of service are not an issue for INSs. Consequently
we define the performance metric 푓푚(⋅) of the ZUPTaided
INS to be a function of the system accuracy and integrity,
that is a function of the state estimate errors and the errors
in the estimate of some statistical dispersion measure (typi
cally covariance) of the accuracy. However, the errors in the
estimates are stochastic variables and consequently we define
the performance 푚 of the system to be the expectation of the
performance metric,
푚 = 피(푓푚(ℰ)) =
∫
푝ℰ(휺)푓푚(휺)푑휺,
(6)
where 피 is the expectation operator, ℰ is the stochastic error
variables, 휺 is the errors, and 푝ℰ(⋅) is the error probability den
sity function. To get a comparable quantity the performance
metric should preferably be a scalar function. This means
that the metric might have to weight errors of different units
relative to each other. The performance metric can be thought
of as a cost function of the errors and as a result it will reflect
how errors of different states and magnitude are valued relative
to each other. This also means that the preferable performance
metric will be dependent on the intended application of the
system.
Together with the performance metric an order relation
[>푚] also need to be defined. In nonmathematical language
this means that we have to define what “better performance”
means in the context of our performance metric. For a scalar
performance metric the order will be either the normal order
of the real field or its inverse.
In practice the distribution 푝ℰ(⋅) is not known and, therefore,
the performance must be estimated from a finite set of samples
from the distributions. To obtain these samples references to
the estimates are needed to calculate the errors. This can often
be achieved for the navigation states but might be difficult for
the dispersion measures. Therefore, the performance metric
would often be chosen as a function of accuracy only and the
integrity checked separately. Further, often references, only
for a subset of the navigational states (e.g. position) and for
a subset of the time instants, are available. However, due to
correlations of state estimate errors this can in many cases be
acceptable.
B. Performance parameterisation
Per definition the performance of a setup will be deter
mined by factors of continuous nature. These factors need
to be parameterised to make a performance characterisa
tion possible. There are many considerations concerning the
parameterisation. Even if preferable, the parameters do not
necessarily need to be specified such that they are given by
numerical values. The important attribute is that the range of
interest of the parameters can be sampled in a sensible manner.
More important is to limit the total number of parameters. As
seen in Section IIIC each parameter adds one dimension to
the final estimation problem. Therefore, only factors believed
to have a significant influence on performance should be kept
in the parameterisation.
The parameterisation will be setup and application depen
dent. As a result no universal parameters can be given. How
ever, some groups of parameters can still be identified. The
parameters can be divided into internal parameters 휃int and
external parameters 휃ext. For an example of a parameterisation
see Section IV.
1) Internal parameters: The internal parameters 휃intcan in
turn be divided into filter parameters and hardware param
eters. The system parameters are normally easily identified
based on the specific models (1)(5) and the fusion filter of
the setup. Considering the complete hardware component
selection as a part of the setup, the parameters, of the
hardware setup, are only the sensor placement/mounting and
sampling speed.
2) External parameters: The external parameters are the
parameters which, given the hardware, determine the trajectory
in a wide sense. The trajectory we define to include both
the path in the navigation state space and the sensor output.
The trajectory is not as easily parameterised as the filter and
the hardware. The true trajectory is not known (no perfect
reference) and difficult to reproduce in a sensible manner.
Further, being a realworld continuous quantity, the trajectory
will not be feasible to parameterise exactly with a finite set
of parameters. Hence, the parameterisation will have to be
done in an approximate manner. That is a limited set of
parameters, describing aspects of the trajectory important to
the performance of the system, has to be found. This can
be justified by treating the human locomotion as a stochastic
process which is parameterised by a finite set of parameters.
C. Estimate performance
Conceptually, there is a large difference between the internal
parameters 휃int and the external parameters 휃ext. In principle
we are always free to adjust the internal parameters while the
external parameters, or rather in this case a probability density
of the parameters, can be given by an intended application.
This means that depending on viewpoint the external param
eters can be both parameters and realisation of a stochastic
variable. In the latter case denote the stochastic variable Θext
and the probability density function 푝Θext(⋅).
Ideally, if the external parameters would span the space of
trajectories perfectly then, without any application given, the
performance (6) of a setup is parameterised by both 휃intand
휃ext,
푚(휃ext,휃int) = 피(푓푚(ℰ);휃ext,휃int),
while the achievable performance of a setup would be param
eterised by 휃extonly,
(7)
푚ach(휃ext) = max
휃int
(피(푓푚(ℰ);휃ext,휃int)),
(8)
where the maxfunction is with respect to [>푚]. On the other
hand, if an application is given, the performance of a setup
Page 4
would be parameterised by 휃intonly,
푚(휃int) = 피휃ext(피(푓푚(ℰ)∣Θext= 휃ext;휃int))
= 피휃ext(푚(휃int,휃ext)),
(9)
while the achievable performance of a setup, given an appli
cation, would not be a function of any parameter,
푚ach= max
휃int
(피휃ext(피(푓푚(ℰ)∣Θext= 휃ext;휃int)))
(푚(휃int)),
= max
휃int
(10)
where once again the maxfunction is with respect to [>푚].
Equations (7)(10) all represent performances but from differ
ent viewpoints. Equations (8) and (10) give performance base
lines and (7),(8), or (9) give performance characterisations.
Note that the evaluation of (10) would contain the evaluation
of (7) and (9) as intermediate steps and that (8) is equivalent
with (10) if 푝Θext(휃ext) = 훿(휃′
delta function and 휃′
Writing out the expectations in (10)
∫
Estimating performance is then a matter of estimating one or
more of the integrals and the maxfunction in (11) depending
on which one of the relations (7), (8), (9), or (10) that is of
interest.
In practice 휃ext does not span the space of trajectories
perfectly, 푝ℰ(휺∣Θext= 휃ext;휃int) is not known, and 푝Θext(휃ext)
is only known roughly for an application. However, assuming
that 휃ext spans important dimensions of the trajectory space
then the inner integral of (11) can be estimated by sampling
푁 trajectories at 푀 points in the external parameter space
with internal parameter settings 휃int. Together with refer
ences this gives error samples 휖푖,푗(휃int) : 푖 = 1,...푁 at
휃ext,푗 : 푗 = 1,...푀. The inner integral (corresponding to
푚(휃ext,푗,휃int)) can then be estimated with the sample mean of
the performance metric [7],
ext−휃ext) where 훿(⋅) is the Dirac
extis the point at which (8) is evaluated.
푚ach= max
휃int
푝Θext(휃ext)
∫
푝ℰ(휺∣Θext= 휃ext;휃int)푓푚(휺)푑휺푑휃ext.
(11)
ˆ 푚(휃ext,푗,휃int) =
1
푁
푁
∑
푖=1
푓푚(휖푖,푗(휃int)).
(12)
The outer integral (corresponding to 푚(휃int)) can then in turn
be estimated by
ˆ 푚(휃int) =
푀
∑
푀
∑
푗=1
푤푗ˆ 푚(휃ext,푗,휃int)
=
푗=1
푤푗
1
푁
푁
∑
푖=1
푓푚(휖푖,푗(휃int))
(13)
where∑푀
Finally, the achievable performance can be estimated by a
푗=1푤푗= 1 and where 푤푗are weights chosen based
on the intended application and approximating 푝Θext(휃ext,j).
numerical evaluation of max휃int(⋅),
ˆ 푚 = ˜
max
휃int
(ˆ 푚(휃int))
= ˜
max
휃int
푀
∑
푗=1
푤푗
1
푁
푁
∑
푖=1
푓푚(휖푖,푗(휃int)),
(14)
where ˜
dim(휃int) ≥ 5. This poses significant difficulties when eval
uating (14) and often make a brute force numerical search in
feasible. A way around this would be to identify well behaved
parameters and groups of parameters with weak interdepen
dencies such that sequential iterative minimisation methods
could be used on a subset of the parameter dimensions. Also,
some parameter dependencies could temporarily be eliminated
with external information, see Section IV and [8].
max(⋅) denotes a numerical approximation of max(⋅).
Unfortunately, (14) is nonlinear with respect to 휃int and
IV. BASELINE SETUP PERFORMANCE EVALUATION AND
CHARACTERISATION
In this section we apply the methodologies given in Sec
tion III to a suggested baseline setup. The performance
characteristics of the zerovelocity detector part is studied in
a companion paper and left out here [8].
A. Baseline setup
To keep the system as simple as possible the sensor model
is taken to be the true value plus noise. The mechaniza
tion equations are first order discretisations of the kinematic
equations with zero order hold assumptions and with the
Coriolis and the Euler terms discarded. The zerovelocity
detector is the SHOE detector [8]. The fusion filter is a
complementary extended KF and the error models are based
on perturbation analysis discarding second and higher order
terms. This baseline filter setup has been chosen since it
represents a common denominator of many studied systems
and also has a low number of parameters making analysis
and presentation of data manageable [1], [3]–[5]. The IMU in
use is a 3DMGX2 from MicroStrain with a dynamic range
of 18[g] and 1200[∘/s] of the accelerometers and gyroscops
respectively. The boots used for the experiment was of the Dr.
Martens Classic model.
B. Performance metric
The performance metric is limited to the reference available.
In the measurement campaign only position reference for
the stop point (closedloop trajectories) was available. The
performance metric was chosen as
푓푚(ℰ) = ∣ℰ푝∣2
(15)
where ℰ푝is the threedimensional error in endposition. The
performance order relation is taken to be the inverse of the
standard real field order, [>푚] ∼ [<]. That is a smaller mean
square error is better.
Page 5
C. Parameterisation
The internal parameters are taken to be the process
and measurement noise covariance matrices within the KF,
cov(nT
3,푘
are taken to be diagonal. The process noise covariance matrix
is assumed to have identical components in the velocity states
휎2
in the orientation states 휎2
the gyroscopes, and zero on the diagonal otherwise. The mea
surement noise covariance matrix is assumed to have identical
components along the diagonal 휎2
훾 and the window length (even though not a parameter in
a strict sense) of the detector ℎ′(⋅) were tuned separately
with a reference based on forcesensitiveresistors as discussed
in [8]. The numerical values used were 훾 = 104.8and window
length 5 samples. No hardware parameters were used in the
presented data. The IMU was mounted in the foot instep and
the sampling speed was 250[Hz]. Hence, for the presented data
휃int= [휎푎,휎휔,휎푣].
In the measurement campaign the trajectory was param
eterised (external parameters) with the walking speed, the
mechanical properties (compressibility/shock absorbtion) and
the topography (roughness/hilliness) of the ground surface, the
path length, some qualitative measures of the appearance of
the path, and the subject itself.
4,푘n4,푘) and cov(n′
Tn′
3,푘). The covariance matrices
푎, modelling measurement errors in the accelerometers, and
휔, modelling measurement errors in
푣. The detection threshold
D. Performance estimation
Trajectories were collected from a matrix of points in the
trajectory parameters space. The performance was estimated
using 2×10 trajectories (error samples) with each 10set taken
over a single powerup cycle. For ease of interpretation the
estimated performance is presented as
√ˆ 푚(⋅)
푠
in which 푠 is the travelled distance. Together with (15) and
(12)(14) this means that the presented performance figures are
the rootmeansquare error (RMSE) of end position normalised
with the path length. The reason for normalising with the path
length is that, under the assumption of small influence of the
heading errors, the RMSE of position will grow approximately
linear with distance. Hence, given that this is true, the nor
malisation will eliminate the dependency on the path length
parameter.
(16)
E. Performance characteristics
Presenting and interpreting ˆ 푚(휃int,휃ext,j) and ˆ 푚(휃int) is
difficult due to dimensionality. However, the 휎푣parameter and
the 휎푎and 휎휔 parameters scale the error covariances of the
inputs relative to each other. Therefore, one might expect that
most of the effect on performance will be captured by the
ratio between 휎푎and 휎휔and the ratio of that ratio to 휎푣. This
has indeed been noted to be the case for values of 휎푣above
approx. 0.001[m/s] and fixating 휎푣 most of the performance
characteristics will be described by the dependence on 휎푎and
휎휔. However, the system does have more than two degrees
휎푣= 0.005[m/s]
√
ˆ 푚(휃int,휃ext,j)
푠
휎휔[∘/푠]
휎푎[m/s2]
102
102
100
100
100
10−1
10−1.5
10−2
10−2
10−2
10−0.5
10−2.5
Fig. 3.
external parameters as given in Section IVE.
√ˆ 푚(휃int,휃ext,j)
푠
(RMSE of postion) for stride period 0.9 [s] and other
of freedom due to coupling with physical quantities in (2) so
one still has to be careful in setting 휎푣. Based on inspection
of ˆ 푚(휃int,휃ext,j) the value is set to 휎푣 = 0.005[m/s]. The
remaining dependence on 휎푎and 휎휔gives a simplified picture
of the performance characteristics but it still contains many
important attributes. Also remember that the coupling with
the detector has been ignored by fixating its parameters. For
a detailed analysis all dependencies have to be considered.
Due to space limitation ˆ 푚(휃ext,j,휃int) is presented only for
a limited number of points in the external parameter space. In
Figs. 35 ˆ 푚(휃ext,j,휃int) is shown over a grid of values of 휎푎and
휎휔. The external parameter points are fast walk (stride period
of 0.9[s]), normal walk (stride period of 1.2[s]), and slow
walk (stride period of 1.8[s]) on flat hard floor in a trajectory
the shape of a digital eight starting and stopping in the
centre. The reason for using a closedloop symmetric trajectory
was to minimise the influence of the heading estimate errors
since it is not observable and easily quantified separately.
This also ensured that the linear error growth assumption
implicitly made in (16) were valid. The radius of the corners
was approximately 1[m] and the trajectory length was about
100[m]. The subject was the same for all presented trajectories:
a male, 1.8[m], and approx. 80[kg]. Figures with equivalent
results for the detection threshold and window length are
available in [8].
In Fig. 6 ˆ 푚(휃int) is given for a grid of 휎푎and 휎휔 values.
Here ˆ 푚(휃ext,j,휃int), based on corresponding trajectories as of
flat hard floor but also for asphalt, gras, and gravel, has
been included and weighted together with equal weights. This
performance characteristic estimate is based on approx 120
trajectories, approx. 100[m] each, taken over a time period of
two weeks. Further comments and analysis of Figs. 36 are
found in the following sections.
Page 6
휎푣= 0.005[m/s]
√
ˆ 푚(휃int,휃ext,j)
푠
휎휔[∘/푠]
휎푎[m/s2]
102
102
100
100
100
10−1
10−1.5
10−2
10−2
10−2
10−0.5
10−2.5
Fig. 4.
external parameters as given in Section IVE.
√ˆ 푚(휃int,휃ext,j)
푠
(RMSE of postion) for stride period 1.2 [s] and other
휎푣= 0.005[m/s]
√
ˆ 푚(휃int,휃ext,j)
푠
휎휔[∘/푠]
휎푎[m/s2]
102
102
100.5
100
100
100
10−1
10−1.5
10−2
10−2
10−2
10−0.5
10−2.5
Fig. 5.
external parameters as given in Section IVE.
√ˆ 푚(휃int,휃ext,j)
푠
(RMSE of postion) for stride period 1.8 [s] and other
F. Achievable performance
The minimum points of the surfaces in Figs. 35 (corre
sponding to ˆ 푚ach(휃ext,j)) are 0.29%, 0.27%, and 0.21% respec
tively. If (16) is changed slightly to include only the horizontal
error the corresponding figures are 0.17%, 0.20%, and 0.10%.
Weighing together all the squared errors equally from the dif
ferent walking speeds on hard flat floor the minima are 0.29%
and 0.19% respectively. The minima of Fig. 6 (corresponding
to ˆ 푚ach) are 0.44% and 0.25% respectively. However, note
that all these values are for a symmetric trajectory in which
the influence of the heading drift is suppressed. One should
be careful when comparing performance figures based on
different trajectory parameterisations and external parameter
weighings. Comparison with other publications in which often
some information about the setup, the performance metric,
and the internal and external parameter settings are missing
휎푣= 0.005[m/s]
√
ˆ 푚(휃int)
푠
휎휔[∘/푠]
휎푎[m/s2]
102
102
100.5
100
100
100
10−1
10−2
10−2
10−2
10−0.5
10−1.5
Fig. 6.
compressibility conditions as given in Section IVE.
√
ˆ 푚(휃int)
푠
for uniform distribution of walking speed and ground
Regions corresponding to operational modes
휎휔[∘/푠]
휎푎[m/s2]
103
103
102
102
101
101
100
100
10−1
10−1
10−2
10−2
10−3
10−3
A
B
C
Fig. 7.
of the ZUPTaided INS. The chart correspond to Fig. 6 seen from above.
Region A correspond to free inertial navigation. Region B and C correspond
to modes in which errors due to the gyroscopes and the accelerometers
respectively are assumed to dominate.
The figure illustrates the regions corresponding to operational modes
is difficult. Also note that the performance estimate figure of
0.44% is dominated by a handful trajectories giving errors in
the range 0.30.9% while most of the trajectories give an error
in the range 00.3%. Our experience is that order of magnitude
20 trajectories recorded at two different occasions are needed
to estimate the performance consistently.
G. Operational modes
In Figs. 46 some more or less flat regions are present.
These are parameter ranges over which the system seems to
behave about the same. That is, they could be interpreted to
correspond to operational modes of the system. Fig. 7 show
the surface in Fig. 6 from above with the regions marked and
labeled. In region A both the process noise covariances of the
accelerometer and the gyroscopes are low in comparison with
Page 7
the measurement noise covariance. This means that we trust
the inertial measurement to a high degree in comparison to
the ZUPTs. This flat area would correspond to free inertial
navigation. In region B the process noise covariance of the
accelerometer is low while the process noise covariance of
the gyroscopes is high in comparison with the measurement
noise covariance. This means that the filter will enforce the
compliance between the ZUPTs and the state estimates by
adjusting the velocity and position but also by adjusting the
orientation which are correlated with velocity and position
with gravity as a lever. The opposite noise magnitude relation
hold for region C. This means that the filter will enforce the
compliance by mainly adjusting the velocity and position.
The performance is typically best in region C but with
symmetric trajectories the performance in region B and C
are comparable. Using nonsymmetric trajectories (results not
shown) the performance in region C is often order of magni
tude worse.
H. Intermediate regions
Between the regions A, B, and C in Fig. 7 there are some
intermediate transition regions. To begin with there is a diag
onal transition region between region B and C. Here a ditch is
often present indicating that the accelerometer and gyroscope
process noise covariances are balanced. Between the region
A and B and between region A and C there are often similar
ditches present. The explanation for this is probably that at
these noise covariance values the filter smoothen out the zero
velocity enforcing over the short retardation period of the foot
impact on the ground. Remember that a fix detector threshold
was used for the data. Then there is a short transition region in
each step from moving to stationary in which the detector will
trigger with increasing probability the closer the foot gets to
stationary. With smooth update, that is balanced accelerometer
process noise values to measurement noise values, this will not
be a problem but with hard updates enforcing zerovelocity
on the first ZUPT this might introduce errors giving a less
satisfactory performance. Detailed analysis will have to verify
this hypothesis. Either way, as seen in Figs. 35 this ditch
is increasingly pronounced for low walking speeds making
the parameter settings more sensitive. This is in agreement
with the poor robustness often observed in the system for
irregular motions patterns, e.g. loitering. These ditches could
be thought of as regions in which time and measurement
error covariance effects are balanced. Finally, occasionally a
minimum is present in the intersecting intermediate regions
between all regions. This minimum corresponds to settings of
which the effects of both the time and all measurement error
covariance settings are balanced.
I. Internal parameter settings
The region, in the 휎푎휎휔parametersubspace in which the
minima corresponding to the achievable performances dis
cussed in subsection IVF were found, is in the intersection
between all the regions A, B, and C with a bias towards or
even inside region C. The numerical values of the components
of 휃int of the achievable performance (0.44%) of Fig. 6 are
휎푎 = 6[m/s2], 휎휔 = 2.5[∘/s], and 휎푣 = 0.005[m/s]. Note:
that the process noise parameters (휎푎and 휎휔) are scaled with
the sampling period in the covariance update equation; The
noise covariances are inflated to take care of all measurement
errors explaining why the values are significantly higher than
typical noise values of MEMS IMUs; Changing 휎푣 will
change the point of the achievable performance in the 휎푎휎휔
subspace; Different IMUs and applications will give different
values. Hence, the given values should be used with caution.
Similar plots as of Figs. 36 should be produced to tune and
determine the achievable performance of a ZUPTaided INS
implementation. Also note that the filter implementation of [8]
is slightly different (no noise covariance scaling with sampling
period) which partially explains the difference in the given
values.
V. CONCLUSIONS
In this article we have suggested a methodology for eval
uating performance of footmounted ZUPTaided INS and
other related systems. General system structure and compo
nents have been identified. Performance of ZUPTaided INSs
and other related systems has been mathematically defined.
Based on this definition and some general division of the
parameterisation, methods for estimating performance have
been derived. Finally, these methods have been applied to a
baseline system setup giving performance characteristics and
achievable performance over a range of internal and external
parameter ranges. The achievable performance has, depending
on external parameters, been estimated to be in the range 0.1
0.44%.
REFERENCES
[1] E. Foxlin, “Pedestrian tracking with shoemounted inertial sensors,” IEEE
Computer graphics and Applications, vol. 1, pp. 38–46, 2005.
[2] L. Ojeda and J. Borenstein, “Nongps navigation for security personnel
and first responders,” J. Navigation, vol. 60, pp. 391–407, 2007.
[3] I. Skog, P. H¨ andel, J. Rantakokko, and J.O. Nilsson, “Zerovelocity
detection — an algorithm evaluation,” IEEE Trans. on Biomedical Engi
neering, vol. 57, no. 11, pp. 2657–2666, 2010.
[4] A. Jim´ enez, F. Seco, J. Prieto, and J. Guevara, “Indoor pedestrian
navigation using ins/ekf framework for yaw drift reduction and a foot
mounted imu,” in Proc. WPNC2010, Dresden, March 2010.
[5] ¨Ozkan Bebek, S. Rajgopal, M. J. Fu, X. Huang, M. C. C. D. J. Young,
M. Mehregany, A. J. van den Bogert, and C. H. Mastrangelo, “Personal
navigation via highresolution gaitcorrected inertial measurement units,”
IEEE Trans. on Instrumentation and Measurement, vol. 59, no. 11,
pp. 3018–3027, 2010.
[6] J. A. Farrell, Aided Navigation. McGrawHill, 2008.
[7] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume 1:
Estimation Theory. Prentice Hall, 1993.
[8] I. Skog, J.O. Nilsson, and P. H¨ andel, “Evaluation of zerovelocity de
tectors for footmounted inertial navigation systems,” in Proc. IPIN2010,
Z¨ urich, Sept 2010.
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