iMAR's Decision Assistant: What is important, if you intend to buy an inertial measurement system.

If you intend to use an inertial measurement system...

... which technical data you should analyze and compare before making your decision

by Dr.-Ing. Edgar v. Hinüber, CEO iMAR Navigation GmbH

Keywords: inertial navigation system, inertial measurement system, inertial measurement unit, attitude heading reference sys-

tem, inertial sensor, gyroscope, accelerometer, angular random walk, bias, drift, free inertial, unaided inertial navigation, GNSS

denied, aided navigation, INS, IMU, IMS, AHRS

Preface

Indeed, it is often very challenging for both inexperienced and advanced

users of inertial technology to make the right decision in an environment

of complex marketing information about which of the various inertial

measurement systems, inertial navigation systems, attitude and heading

reference systems, inertial measurement units, or at least inertial sen-

sors on the market best and most economically meets their require-

ments.

With this article, we aim to help the reader better understand the physics

behind inertial navigation or inertial measurement systems and sensors,

and to evaluate the information. We also aim to enable you to better val-

idate the datasheets provided by suppliers, identify inconsistencies that

are unfortunately often present, and find your best technical and eco-

nomic solution. Only in this way can you be sure that the product you select truly meets your requirements.

Tatsächlich ist es oft sehr schwierig, sowohl für unerfahrene als auch für fortgeschrittene Benut-

zer von Trägheitstechnologie, im Umfeld vielschichtiger Marketing-Informationen die richtige Ent-

scheidung zu treffen, welches der verschiedenen Trägheitsmesssysteme, Trägheitsnavigations-

systeme, Lage-Kurs-Referenzsysteme, Trägheitsmessgeräte oder zumindest Trägheitssensoren

auf dem Markt am besten und wirtschaftlichsten ihren Anforderungen entspricht.

Mit diesem Artikel helfen wir dem Leser, die Physik hinter der Trägheitsnavigation oder den Träg-

heitsmesssystemen und Sensoren besser zu verstehen und die Informationen zu bewerten. Wir

versuchen auch, Sie besser in die Lage zu versetzen, die Datenblätter der Anbieter selbst zu

validieren, leider oft vorhandene Inkonsistenzen zu identifizieren und Ihre beste technische und

wirtschaftliche Lösung zu finden. Nur so können Sie sicher sein, dass das von Ihnen ausgewählte

Produkt tatsächlich Ihren Anforderungen genügt.

Introduction into Inertial Measurement Technology:

Inertial navigation and guidance systems were initially developed for rocket guidance and control. Today,

their applications span a wide range of fields, from horizontal directional drilling deep underground to

spacecraft navigation. In fact, inertial technology has become an integral part of everyday life. For exam-

ple, every modern car is equipped with at least one gyroscope and two accelerometers for the Electronic

Stability Program (ESP) or airbag control, ensuring safe travel even in challenging conditions. Likewise,

every smartphone incorporates accelerometers, gyroscopes, a GNSS receiver, and a magnetometer.

A typical Inertial Navigation System (INS) relies on gyroscopes (angular rate sensors) and accelerome-

ters as its primary sensors. Gyroscopes are used to determine the vehicle's orientation, compensating

for gravitational effects on the accelerometer data. This process involves solving a complex set of differ-

ential equations in real-time to convert the sensor measurements into estimates of velocity, position,

attitude, and heading, based on a known initial position in latitude and longitude.

Modern Inertial Navigation Systems (INS) commonly utilize 'strap-down' technology, where all inertial

sensors (gyroscopes and accelerometers) are rigidly mounted to the vehicle. In earlier designs, 'gimbal'

technology was used, with gyroscopes mechanically stabilizing accelerometers in space. In strap-down

systems, stabilization is achieved through mathematical calculations, subjecting all inertial sensors to the

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vehicle's full dynamic range. Despite the absence of mechanical gimbals, strap-down systems are signif-

icantly more robust operationally than gimbaled systems, although they demand higher sensor range,

scale factor accuracy, and sensor durability.

All unaided inertial navigation systems experience drift over time, as small measurement errors accumu-

late, resulting in progressively larger errors in velocity and, especially, position due to double integration

over time. The methods for compensating and correcting this drift, particularly in real-time applications,

differ substantially across market solutions. Only suppliers who excel in providing unaided inertial navi-

gation with the highest performance — especially under challenging environmental conditions, often re-

garded as the 'king class of inertial measurement technology' — are capable of delivering compelling

solutions for aided navigation scenarios as well.

Control theory, particularly Kalman filter-based techniques, provides a framework for integrating comple-

mentary data from various sensors, a process known as sensor data fusion. Common supplementary

sensors used to support INS-based systems include satellite navigation systems like GPS, GALILEO,

BeiDou and GLONASS (GNSS), as well as odometers, air data sensors, magnetometers, radio position-

ing systems, and more. Additionally, specific techniques such as Zero Velocity Update (ZUPT) and Posi-

tion Update (PUPT) can enhance accuracy for particular applications. (Link )

The significant risks of other signal processing methods, such as AI-based approaches, which are

often greatly underestimated by inexperienced users, are discussed in a dedicated chapter of this paper,

particularly regarding their use not only in safety-critical or reference measurement applications.

Trägheitsnavigations- und -führungssysteme wurden ursprünglich zur Steuerung von Raketen

entwickelt. Heutzutage werden sie in vielen Anwendungen eingesetzt, von der horizontalen Rich-

tungsbohrtechnik tief unter der Erdoberfläche bis zur Navigation von Raumfahrzeugen. Heutzu-

tage kommt jeder täglich mit Trägheitstechnologie in Kontakt: Zum Beispiel enthält jedes mo-

derne Auto mindestens ein Gyroskop und zwei Beschleunigungssensoren für das ESP (elektro-

nisches Stabilitätsprogramm) oder für die Airbag-Steuerung, um das Reisen auch in schwierigen

Umgebungen so sicher wie möglich zu machen. Auch jedes Smartphone enthält heute Beschleu-

nigungssensoren, Gyroskope sowie einen GNSS-Empfänger und ein Magnetometer.

Ein typisches Trägheitsnavigationssystem (INS, inertial

navigation system) verwendet als Sensoren Gyroskope

(Drehratensensoren) und Beschleunigungssensoren.

Die Gyroskope werden dabei verwendet, um die Orien-

tierung des Fahrzeugs zu bestimmen und insbesondere

auch, um die Messdaten der Beschleunigungssensoren

in Bezug auf die Schwerkraft zu kompensieren. Das be-

deutet, eine große Menge an Differentialgleichungen in

Echtzeit zu lösen, um diese Messwerte in Schätzungen

von Geschwindigkeiten, Position, Lage und Kurs umzu-

wandeln, ausgehend von einer bekannten Anfangsposi-

tion in Breiten- und Längengrad.

Die heutige Implementierung von Trägheitsnavigations-

systemen (INS) erfolgt in der sogenannten "strap-down"-Technologie, bei der alle Trägheits-

sensoren (Gyroskope und Beschleunigungssensoren) steif am Fahrzeug montiert sind. In der

Vergangenheit wurden die Systeme in der sogenannten "gimbal"-Technologie entworfen, bei der

die Gyroskope verwendet wurden, um die Beschleunigungssensoren mechanisch im Raum zu

stabilisieren. In strap-down-Systemen erfolgt die Stabilisierung mathematisch, und daher sind

alle Trägheitssensoren den vollen Fahrzeugdynamiken ausgesetzt. Aufgrund fehlender mecha-

nischer Gimbals sind die strap-down-Systeme im Betrieb viel robuster als die gimballed Systeme,

aber die Anforderungen an den Messbereich, die Skalenfaktorgenauigkeit und die Robustheit der

Sensoren sind entsprechend höher.

Alle ungestützten Trägheitsnavigationssysteme leiden aufgrund der erforderlichen mathemati-

schen Integration von Drehraten und Beschleunigungen zur Bestimmung der Lagewinkel und

Position unter einer zeitabhängigen Drift, weil kleine Fehler in den Messungen zu progressiv grö-

ßeren Fehlern in Geschwindigkeit und insbesondere Position aufgrund der doppelten Integration

über der Zeit führen. In der Kompensation und Korrektur dieser Drift insbesondere in

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Echtzeitanwendungen unterscheiden sich die am Markt angebotenen Lösungen ganz erheblich.

Nur wer als Systemlieferant die ungestützte Trägheitsnavigation (free inertial navigation, unaided

navigation) als „Königsklasse der Inertialmesstechnik“ in schwierigen Umgebungsbedingungen

führend beherrscht und anbieten kann, der kann auch für gestützte Navigationslösungen (aided

navigation) überzeugende Lösungen liefern.

Regelungstechnik im Allgemeinen und insbesondere Kalman-Filter basierte Verfahren bieten den

Rahmen für die Kombination von Informationen aus verschiedenen komplementären Sensoren

– die sogenannte Sensordatenfusion. Die hierfür am häufigsten ergänzenden Sensoren, die zur

Stützung INS-basierter Systeme verwendet werden, sind Satellitennavigationssysteme wie GPS,

GALILEO, GLONASS, …(GNSS), Odometer, Luftdatensensoren, Magnetometer, Funkortungs-

systeme usw. Des weiteren erlauben besondere Methoden wie ZUPT, PUPT (Zero Velocity Up-

date, Position Update) usw. anwendungsspezifische Genauigkeitsverbesserungen. (Link )

Die signifikanten und von unerfahrenen Anwendern zumeist deutlich unterschätzen Risiken

anderer Signalverarbeitungsmethoden wie KI basierter Verfahren für den Einsatz nicht nur in

sicherheitsrelevanten oder Referenzmesstechnik-Anwendungen werden in einem eigenen Kapi-

tel in dieser Abhandlung erörtert.

The right INS for your Application:

It is a big difference to operate an inertial measurement

system in static lab conditions or low dynamic environment or in the "real-world".

Check the performance of the IMS (IMS = inertial measurement system) for the envi-

ronment you want to operate the system in. Link

•

Will it be used on an aircraft (transportation aircraft, helicopter, drone or

fighter?),

•

or on a rail vehicle (surface or underground?),

or on a passenger car or a truck or a tank,

or on a naval ship, a ferry or a speed boat or on an underwater surveying

vehicle,

•

or inside of a missile or a torpedo,

or will it be used e.g. in a drilling application or in pipeline surveying or for

machinery guidance,

•

or will it be used e.g. to acquire the field of gravity with high accuracy?

To support your needs as best as possible, you can send us the Inquiry Form

from our web site, filled with your application related information:

https://www.imar-navigation.de/downloads/faq/enquiry_imar.docx or

https://www.imar-navigation.de/downloads/faq/enquiry_imar.pdf

Compare the conditions of operation given in the data sheet of the system intended

to be used: Is the condition well defined and will it meet your application requirements?

•

Will GNSS be available in your application in the way as it is assumed inside

the data sheets of the systems you are investigating?

Do you require operation also in GNSS denied environment, e.g. under jam-

ming or spoofing impacts? Is the solution, described in the datasheet, able

to handle operation in such GNSS denied environment?

•

What is the behavior of the system under coning motion, under vibration and

under temperature gradients?

What operation mode is required for your application and is the advertised

solution able to comply? See the next chapters of this paper regarding free

inertial navigation, pure inertial navigation, aided navigation, surveying,

ZUPT and PUPT aiding, ...)

•

Do you need accurate, reliable and available results of the system during

your data sensitive or safety critical missions or anywhere, where you have

to rely on the data output? Then any AI based solution might not be the right

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choice even if it might be slightly cheaper in purchasing. Cost should be

compared in case of a failed mission due to an unpredicted AI decision.

Sensor Technology Selection and Sensor Data Fusion:

Each inertial sensor

technology has its specific advantages and drawbacks which have to be considered

regarding the foreseen application and desired accuracy. Some sensor technologies

come e.g. with a very high stability of sensor performance (e.g. ring laser gyros) while

others are for instance optimized for very light weight or low cost, but being affected

by possible accuracy aging effects (like MEMS based sensors).

Inertial Sensors: Take into consideration that MEMS based gyros (working on Cori-

olis law using vibratory excitation) as well as spinning dynamical tuned gyros (DTG)

show a so-called g-dependent drift, i.e. they produce a drift (angular rate offset) de-

pendent on linear and quad-

ratic acceleration and environ-

mental vibration impacts. High

performance ring laser gyros

(RLG = ring laser gyros) and

hemispherical resonator gyro-

scopes (HRG) as well as mid

performance fiber optical gy-

ros (FOG) do not show such

GNSS

pos & vel & stddev

& raw data

Output

pos, vel, attitude, hea-

ding, rates, accels, std-

devs, time, status, BIT,

raw data etc.

ODO/VMS

pulses / CAN

Ext Aid

Accel / Rate

6 axes raw data

temperatures

pos / vel / mag / air /

DVL / LiDAR…, stddev,

time stamp

g-dependent

drift,

while

higher performance fiber optical gyros (FOG) also show performance degradation due

to physical reasons, caused by vibration impacts and temperature gradients.

Sensor Data Fusion: The signal processing on system level (“sensor data fusion”)

has to take care for all sensor errors. Therefore, the iMAR sensor data fusion is able

e.g. not only to estimate the common inertial sensor offsets, but also estimates and

compensates the scale factor drifts, misalignments and other effects in real-time

(more than 40 states are estimated, compared to the classical and most common

implementations of competitors with only 15 states). (Link )

With over 30 years of experience in sensor data fusion and integration, iMAR

incorporates all state-of-the-art gyro technologies and performance classes in its sys-

tems, ranging from MEMS to FOG, RLG, and HRG, depending on the application

requirements. The company utilizes a robust real-time sensor data fusion process with

more than 40 states to estimate and compensate for most residual errors and even

the aging effects of inertial sensors. Link

Additional complementary sensors can also be integrated into the sensor data fusion

process, such as GNSS (single and dual antenna), wheel sensor data (odometer,

VMS), DVL (Doppler Velocity Log), EM-Log, magnetometer data (magnetic heading

— though caution is advised with these sensors, as they are highly sensitive to envi-

ronmental influences that cannot be compensated for if they change during the mis-

sion), air data sensors, and more.Link

The physics underlying the mathematics of inertial navigation is, among other things,

described by Newton's axioms. While the fundamental mathematical framework and

solutions to navigation equations have been well known for many decades, the real

challenge lies in implementing these solutions in a robust, efficient, highly reliable,

and readily available manner. Achieving this requires a vast amount of experiential

knowledge, which the iMAR team has accumulated over more than 30 years across

hundreds of different applications.

In the absence of such expertise, less experienced providers of PNT solutions may

find it tempting to turn to so-called artificial intelligence (AI) methods, as these can

quickly demonstrate promising results in a well-trained environment. AI techniques

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are well-established in fields like image, speech, and video processing, particularly

where the physical modeling of phenomena remains challenging for engineers and

scientists. However, because the learning processes of AI agents are only sporadi-

cally verifiable, there is a consensus among experienced users and experts that AI-

based systems should not be employed in mission-critical or safety-relevant applica-

tions. The real behavior of these systems cannot be entirely predicted, posing a sig-

nificant risk to the mission.

At iMAR, we firmly believe that only physics has to determine the behavior of our

measurement systems. Our deterministic real-time results arise from the intelligent

signal processing conducted by our experienced engineers and scientists, not from

opaque AI. Our customers choose our systems because they value the exceptional

reliability, availability, and accuracy of our solutions, even in critical missions during

daily operations. We achieve this through our mathematically and physically pre-

cise algorithms — and intentionally not through AI (artificial intelligence) interpreted

so-called “measurement results”.

AI methods are utilized in our work only in areas where they can contribute, such as

object detection or classification, and do not have any safety-critical implications.

Die Physik, der die Mathematik der Inertialnavigation folgt, wird u.a. durch die

Newton’schen Axiome beschrieben. Während die grundsätzliche mathematische

Beschreibung und Lösung der Navigationsgleichungen seit vielen Jahrzehnten

allgemein bekannt ist, liegt die besondere Herausforderung darin, die Lösung

robust, effizient, hochgradig zuverlässig und verfügbar zu realisieren. Hierzu be-

darf es eines enormen Erfahrungswissens, welches sich das iMAR-Team in über

30 Jahren in hunderten verschiedener Anwendungen erarbeitet hat und täglich

weltweit demonstriert.

Kann man auf ein solches Wissen nicht aufsetzen, erscheint es für einen Anbi-

eter von PNT-Lösungen auf den erste Blick sehr attraktiv, auf Metoden der sog.

künstlichen Intelligenz (KI) zu ersetzen, denn hiermit kann er im trainierten

Umfeld recht schnell passable Lösungen vorzeigen. KI-Verfahren sind bestens

in Bereichen der Bild-, Sprach- und Videomanipulation etabliert und insb. dort,

wo eine physikalische Beschreibung von Sachverhalten den Anwendern heute

noch schwer fällt. Da der Lernprozess solcher KI-Agenten jedoch nur sporadisch

prüfbar ist, ist es in erfahrenen Anwenderkreisen Konsenz, dass auf KI-

Methoden basierte Systeme nicht in Daten- oder sicherheitsrelevanten Anwen-

dungen zum Einsatz kommen, da das reale Verhalten derartiger Systeme nicht

hinreichend voraussagbar ist, womit ein enormes Sicherheitsrisiko gegeben sein

kann.

Deshalb gilt bei iMAR: Nur die Physik bestimmt das Verhalten unserer Mess-

systeme. Unsere deterministischen Echtzeitergebnisse entstehen durch die in-

telligente Signalverareitung unserer projekterfahrenen Ingenieure und Wissen-

schaftler, und nicht durch intransparente KI. Unsere Kunden verwenden un-

sere Systeme, denn sie schätzen die außerordentliche Zuverlässigkeit, Ver-

fügbarkeit und Genauigkeit unserer Lösungen auch in kritischen Missionen im

täglichen Einsatz. Dies erreichen wir durch unsere mathematisch-physikalisch

präzisen Algorithmen - und ganz bewusst nicht durch AI (artificial inteligence)

interpretierte sogenanten "Messergebnissen".

AI-Methoden kommen bei uns nur dort zum Einsatz, wo diese z.B. bei Ob-

jekterkennung oder Klassifizierung einen Beitrag liefern können abr keinen

sicherheitsrelevanten Einfluss haben.

Gyro Bias:

If the inertial system operates unaided (without odometer/velocity or GNSS or mag-

netometer aiding or similar), the gyro bias indicates the increase of the angular error

over time (in deg/h or deg/s). If the system is aided with speed information (e.g.

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odometer / wheel sensor or Doppler log), the roll and pitch gyro drift can be compen-

sated in the measurement system by sensor data fusion and the gyro drift mainly

affects the heading accuracy over time. If the system consists of low drift gyros, also

the true heading can be estimated using gravity and earth rate information (so-called

north-seeking or gyro compassing).

If the system is aided with position information (e.g. GPS or GALILEO or GLONASS

or LiDAR etc.), also the heading drift can be corrected and true heading can be ob-

tained (even with medium grade performance gyros), if the applied motion dynamics

is sufficient, i.e. if the heading state is observable in the Kalman filter¹. But of course

the smaller the gyro drift the better all possible angular corrections and the longer the

allowed time where the aiding information may be not present (e.g. GPS in urban

canyons)!

If the system is operated in free inertial navigation mode, the gyro bias is responsible

for the position and velocity error over time (so-called Schuler oscillation).

Gyro Scale Factor Error:

This is an indication of the angular error which occurs during ro-

tation. E.g. with 300 ppm scale factor error (=0.03%) the angular error is in the area

of 0.1 degree after a one revolution turn. With a ring laser gyro or hemispherical res-

onator gyro system with < 10 ppm scale factor error the angular error is less than 1

arcsec (0.0003 deg) if the rotation angle is 30 deg.

Misalignment:

A misalignment between the gyro axes (or accelerometer axes) causes a cross-

coupling between the measurement axes. A misalignment of 0.1 mrad inside of the

system (e.g. residual calibration mismatch) leads to a roll error of 0.036 degree during

a one revolution turn around the yaw axis (if the system is unaided). The smaller the

required misalignment, the higher the requirements to sensor performance and cali-

bration equipment (e.g. iMAR's multi-axes turn-tables).

Accelerometer Offset:

An

offset in an accel-

erometer intro-

duces an error

during alignment,

specifically in the

determination of

the initial roll and

pitch angles, as it

directly affects the

accuracy of meas-

uring gravity (ap-

proximately 9.81

m/s²). For in-

stance, an offset

of 0.1 mg results

in an angular error

of about 0.006 de-

grees in either

pitch or roll (0.1

mg = g × sin(0.006

deg)). These sen-

sor offsets can be

estimated during

operation through

15 °/h/sqrt(Hz) resp. 0.25 °/sqrt(hr)

0.8 °/hr

Allan Variance of a gyro

1

Observability means, that the sensor data fusion has enough information available to estimate certain states like gyro bias or

heading. Example: If an aircraft flies always straight forward at constant speed, it is impossible to estimate vertical gyro bias or

heading using a single antenna GNSS aiding, because due to the mentioned motion no significant acceleration or angular rate

will be measured.

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the system’s integrated Kalman filter data fusion, utilizing GPS, DGPS, RTK data, or

the Zero Velocity Update Procedure (ZUPT), provided there is sufficient motion dy-

namics.

Bandwidth:

In general, the dynamic performance of an inertial measurement system (IMS) im-

proves with higher internal sampling rate and bandwidth of the inertial sensors. Proper

internal data synchronization (time stamping) is also essential for accurate signal pro-

cessing, especially when the IMS operates in challenging dynamic environments. A

high-precision internal time reference and hardware-based time stamping for all data

are crucial for ensuring reliable performance in an INS. Furthermore, low latency in

data output is mandatory for utilizing an INS in trajectory or attitude control applica-

tions, such as those involving autonomous vehicles.

Gyro Random Walk:

This value, expressed in deg/sqrt(hr), represents the noise of the gyro

used. A larger value indicates more noise in the measured angular rates and angles.

Some manufacturers also specify this as noise density in deg/h/sqrt(Hz). Both values

are equivalent for white noise gyro output; dividing the second value by 60 converts it

to deg/sqrt(hr). An angular random walk (ARW) of 0.003 deg/sqrt(hr) suggests that

the angular error (uncertainty) due to random walk is approximately 0.001 deg after 6

minutes (unaided) or 0.0004 deg after 1 minute (all values reported as one sigma).

The angular random walk is crucial for the accuracy of north-seeking, as halving the

random walk reduces the time required for north-seeking by a factor of four, provided

the gyro’s resolution is sufficiently high.

The accompanying plot of the Allan Variance for a mid-performance gyro graphically

illustrates the square-root ARW of a MEMS gyro (to obtain the ARW in [deg/sqrt(hr)],

take the value at 1 second and divide it by sixty). At 1 second, the square-root of the

Allan Variance is 15 deg/hr. This yields an Angular Random Walk (ARW) of 15/60

deg/sqrt(hr) = 0.25 deg/sqrt(hr) = 0.0042 deg/s/sqrt(Hz) = 15 deg/hr/sqrt(Hz) (assum-

ing white gyro noise). The bias stability, indicated by the minimum point of the graph,

is 0.8 deg/hr at a correlation time of 3,000 seconds. Overall, this demonstrates that

we are utilizing a relatively high-quality MEMS gyro.

Position error of an unaided, free inertial INS:

We must distinguish between short-term

accuracy and long-term accuracy in an inertial navigation system (INS). Additionally,

it's important to differentiate between arbitrary moving objects, such as aircraft, ships,

or spacecraft, and land-based vehicles that travel on roads — applications character-

ized by specific motion constraints.

Long-time accuracy of an arbitrary moving, unaided, free inertial

INS:

Definition: An arbitrary moving unaided free inertial INS operates in a mode devoid

of any external aids, meaning there is no GNSS, magnetometer, air data, Doppler log,

LiDAR, RF positioning, or ZUPT. In this mode, the INS can move without limitations,

provided it remains within the measurement range of the inertial sensors.

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In this context, the system experiences a position error known as Schuler oscillation.

This position error, typically measured in nautical miles per hour (nm/hr), reflects the

global position error of the free inertial INS due to residual accelerometer and gyro

errors. The oscillation occurs with a period of approximately 84 minutes, as well as

with a 24-hour cycle. The amplitude of the oscillation is influenced by the accelerom-

eter offset, while the average position drift, or 'shift,' is affected by gyro drift. This is a

simplified model for explanatory purposes; further details can be derived from the in-

ertial differential equations.

The figure shows such long time behavior of a free inertial navigation (example: data

obtained from iNAT-RQT over more than 3 days): Link

This Schuler Oscillation plot displays position error in meters and time in hours. For

example, the free inertial INS shows a position error of 3 km after 70 hours (equivalent

to 0.02 nm/hr)

As illustrated in the plot, it is crucial to clarify how the value of 'free inertial drift' is

derived. Due to the 24-hour oscillation, you can observe that the position error after

11 hours is identical to that after 70 hours. The conditions of data acquisition also play

a significant role: this plot was generated following only 10 minutes of initial alignment.

Why do some vendors claim much lower free inertial drifts?

If the INS is aided prior to drift determination (for instance, by operating it with signifi-

cant motion dynamics and external aids like GNSS), and if the system is aided by an

EM-log (example: naval vessels, submarines), it is possible to achieve drift values

below 1 nm per 100 hours, or even over 360 hours. However, it is essential to note

that this scenario does not represent 'pure inertial, unaided' operation, as the INS

requires adequate position aiding for a substantial duration (e.g., 12 hours) and at

least periodical velocity aiding to provide such results. Many datasheets, however, do

not adequately explain this requirement, nor do they mention that these systems need

to be temperature-controlled and require significant time for power-up.

Short-time accuracy of an arbitrary moved unaided INS (free iner-

tial navigation):

Definition: A free inertial operating INS functions in a mode devoid of any external

aidings, meaning no GNSS, magnetometer, air data, Doppler log, LiDAR, RF posi-

tioning, or other assistance. Short-term operation refers to a duration that is signifi-

cantly shorter than the Schuler period of 84 minutes (as previously mentioned).

In this operational mode, the values (expressed in meters or meters per second) are

relevant for measurements lasting less than approximately 20 to 40 minutes, as

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Schuler oscillation is not significant for short-term measurements. An accelerometer

offset results in a position error that increases quadratically over time.

delta_s = 0.5 x delta_a x T²

[m]

(a)

with delta_a = accelerometer offset [m/s²] and T = measuring time [s].

Example for a medium accurate system:

delta_a = 1 mg  0.01 m/s², T = 100 sec ➔ delta_s = 50 m

The gyro drift delta_omega affects the position error corresponding to the equation

delta_s = g/6 x delta_omega x T³ [m] (b)

with delta_omega in [rad/s] and g = 9.81 m/s² .

An attitude (roll/pitch) error of e.g. delta_attitude affects the position error due to a

wrong compensation of the gravity on the horizontal IMS axes:

delta_s = 0.5 x g x sin (delta_attitude) x T²

[m]

(c)

Example, how you can validate manufacturer’s statements

(with data from a vendor’s datasheet): IXSEA LANDINS

If a provider promotes an inertial measurement system (IMS) with a roll/pitch accuracy

of 0.005 degrees and claims a horizontal position error of 0.7 m (and a vertical position

error of only 0.5 m) after 300 seconds in free inertial navigation mode—without odom-

eter aiding, without ZUPT, and without internal vibration isolators—you can easily ver-

ify and calculate two key factors using the simple thumb rule equations provided

above:

•

Position error due to 0.005 deg roll or pitch error after 300 sec (free inertial):

0.5 x 9.81 m/s² x sin(0.005°) x (300 sec)² = 38 m (from equation (c))

•

What must be the accelerometer accuracy to achieve 0.7 m after

300 sec (free inertial)?

0.7 m / (0.5 x (300 sec)²) = 1.5 µg (!!) absolute accuracy over 300 sec

(from equ. (a))

The simple calculations reveal a discrepancy in the reported performance data; either

the position error must be significantly worse, or the attitude error must be much

smaller to achieve the advertised specifications. For context, an absolute accuracy of

1.5 µg in accelerometer bias approaches gravimeter accuracy, yet such reliability is

typically not available in industrial or military land navigation systems. It's important to

note that gravity itself changes by approximately 0.3 µg for every meter of elevation!

Position error of an unaided, pure inertial INS on road vehicles

(taking only into account motion specific constraints):

Long-time accuracy of an pure inertial INS without ZUPT aiding:

Definition: The INS is operated on a land vehicle driving on a road or off-road. The

vehicle has no capability to fly or to swim – this we call “motion constraints”. The

vehicle has sufficient grip on the surface. No external aiding is available, i.e. no

GNSS, no wheel sensor (odometer), no magnetometer, no LiDAR, no RF position-

ing etc. Over long duration and distance no ZUPT or PUPT shall be required.

Unaided Road and Outdoor Navigation:

Condition: No GNSS, no odometer, no RF aiding, no magnetometer aiding - but



 iMAR

10/2024 rev. 2.51 DocNo.: DOC181020012 technical modifications reserved w/o notice

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