Inertial Sensors and Systems

An Introduction

Prepared by :

The Engineering Department,

GeneSys Elektronik GmbH,

July 1997


| Introduction | Sensors | Gyroscopes | Accelerometers | Inclinometers | Systems | Strap-Down-Systems | Drift | System internal augmentation | System external augmentation | IMU | AHRS | INS | Conclusion |




Inertial sensors are used in applications where rotational and linear movements are to be measured without reference to external coordinates. These movements can be measured by gyroscopes and accelerometers. A major user of such sensors and systems is Aviation, with its widespread use of artificial horizon and other navigational systems.

In recent years there has been a growing interest in industrial applications of inertial systems. This paper provides an introduction to the subject.






Gyroscope (or gyros) measure rotational values without reference to external coordinates. Most gyros measure the speed of rotation (also known as ‘rates’) in one axis and are known as ‘single axis gyros’. Speed of rotation is normally measured in units of degree per second or hour (/sec or /h).

The operating principle of these sensors can be split into two groups :

- Mechanical

- Optical

Most people would remember the classic rotating disc gyroscope, which belongs to the mechanical group. One of today’s high precision gyros is still the mechanical, rotating gyro; however it is dependent on linear accelerations because of its mechanical measuring principle.

The optical group comprises the fibre optic- and laser types. These use the Sagnac Effect (named after its French discoverer), which, when the sensor is turned, results in a difference in transit time between two light waves passing through the same optical path but in opposite directions.

Optical gyros therefore do not depend on acceleration as gyros in the mechanical group. This is one of their biggest advantages, and for this reason GeneSys prefers to use this type of gyroscope in its standard mid-range systems (for high dynamic performance applications).


A gyro’s accuracy and performance can be judged by its output errors; for rate sensors the most important are :

The bias (/sec or /h) is the offset error output; it can be measured when the sensor is static.

The scale factor error is the linear deviation of the measured rate from the true rate (normally given as a percentage of full scale).

Noise is highly important because, after signal integration, the noise results in non-deterministic behaviour (known as ‘Random-Walk’ behaviour). The unit of measurement generally used for this is /.

It should be noted that all these errors are temperature dependant.





Inertial measurement of linear acceleration is possible with accelerometers. The signal is obtained mechanically and scaled in g [1g = 9,81m/sec2] or
milli g (mg).


As for gyroscopes the most important output errors are :

Accelerometer bias is measured in units of mg.



With some inertial systems only orientation is needed, not position. For reasons explained later, the accelerometer can be replaced by an inclinometer.

Inclinometers are sensors which measure the angle to the horizon. Most use fluids to detect tilt; some use micromachined pendulums.




Generally speaking, inertial systems measure motions around three orthogonal axes, and there are two accepted methods of achieving this.

-Gimballed Platform

-Strap Down System

The gimballed platform allows the angles to be obtained from the structure frame supporting the mechanical gyro.

The Strap Down System has three single axis gyros mounted orthogonally. These three gyros are fixed mechanically, or said to be "strapped down", to a body (aircraft,ship,car) and therefore measure the body movements (also called body data). The data being measured is therefore body related.
The angles relative to horizon and north are obtained by mathematical methods.

GeneSys has concentrated on the Strap Down System, because the system can be adapted to the customer’s application by selecting from the large range of gyros on the market. In general terms this means that the GeneSys designed processor and algorithms often stay the same, only the sensors need to be changed.

Furthermore, the system’s performance can be enhanced by using accurate test calibration data.





If three single-axis-gyros together with three accelerometers are mounted on X,Y and Z axes (‘orthogonal mounting’), then these six sensors comprise a clockwise rotating, Cartesian, sensor co-ordinate system.
In order to read the angles related to the earth’s surface, the sensor data is transformed mathematically from the sensor co-ordinate system so as to relate to the three dimensional reference co-ordinate system of the earth.

It should be emphasized that single-axis measurements from one gyro with accelerometers or inclinometer will also show precise instantaneous values; however applying this approach to two or three independent orthogonal subsystems will result in errors.
This is because independent integration of three sensor signals over time will lead to ambiguous states of orientation.

Example of ambiguity :

Imagine an aircraft flying horizontally northwards has to execute two manoeuvres; it is to turn 90 around both its Pitch and Roll axes; these movements are to be executed consecutively.
First of all it turns around its Roll axis, as a result the aircraft will have its right wing pointing downwards. Then it is to turn around its Pitch axis; that is to say that with the right wing still pointing down, the aircraft will now have changed its heading to the east.

Now change the sequence of manoeuvres from its starting position, i.e. it turns around its Pitch axis, and the aircraft will now climb vertically with its wings pointing east/west. Turning around the Roll axis leaves the plane still climbing vertically but rotated by 90 and with its wings pointing north/south.

Independent integration over time with three gyros will correctly show Pitch and Roll turned by 90 degrees. However two end-states of orientation are possible depending on the order of execution of the manoeuvres.

Similarly, if mathematical transformation of the signals is incorrectly executed then errors will also occur.




Imagine such a gyro system being exactly aligned to the horizon and to north. In a perfect world this system, whilst in motion, would be able to calculate, from its inertial sensors, the present orientation and position, assuming the starting co-ordinates are known.
In other words, the sensors measure rates and acceleration, which are integrated over time to relative angles and positions; these are then transformed mathematically. With known starting co-ordinates and the transformed sensor signals true orientation and position can then be calculated.

Alas, there is no perfect world; gyros and accelerometers have an offset, known as bias. The gyro bias for instance, shows itself after integration as an angular drift, increasing rise linearly over time (ramp).

What happens to the accelerometer signal is even worse. Following double integration so as to obtain position the sensor bias now gives rise to a positional drift error raised to the power of 2. The signals with their drifts are transformed and so the drifts are also present in the reference co-ordinate system. This means that the errors of orientation and position are increasing proportionally with the elapsed time.
Fortunately, there are methods to compensate for drifts of this nature.

System internal augmentation

To compensate for sensor drift, a process called augmentation is used, whereby gyro and acceleration errors are compensated by utilising other system states. For instance the angular drift error of a gyro after integration can be compensated, if true angular measurements are taken. Comparing the two shows then the gyro drift.

Assuming that inertial systems are generally operated around an equilibrium position and thus linear acceleration integrates to zero over time, then attitude can be processed via gravitational acceleration, thus giving true angular measurements to compensate gyro drift.

If determination of position is not needed, then inclinometers can be used
(inclinometers are cheaper and provide the same or even better degree of angular resolution).

This can be interpreted as follows :

The gyros are responsible for measuring the dynamic part of system motion, while the inclinometers give us, over a longer period, a precise absolute measurement of attitude. Combining these measurements makes precise attitude information available (in other words, the drift is now observable).
The heading (azimuth) can not be compensated with this method, because the acceleration vector of gravity is parallel to the Yaw-axis of the reference co-ordinate system.

With earth-fixed or low-dynamic systems (gyro compass) geographical north can be found via the earths rotation. For this application high-quality gyros are recommended where the bias is typically less then 0.1 to 0.01 /h.



System external augmentation

If other sensors are integrated with the basic 6 sensors so as to provide additional information such as speed, heading and position (odometer, speedometer, magnetic compass, Global Positioning System (GPS), then augmentation can be extended. This normally improves the system’s performance.

Errors due to augmentation

It should be noted that the above method to compensate for drift is not error free; for reasons that are system inherent other error sources will arise.
One good example of augmentation errors is again provided by aviation. If an aircraft and its navigation system is circling for a extended period, it will not only detect gravitational acceleration, but also centrifugal forces, resulting in incorrect attitude information, because both vectors add up to a new, false vector. Inclinometers will not recognise this as being incorrect information. Accelerometers will detect an increasing length of the vector, but the vector is perpendicular to the aircraft’s attitude-plane.
This means that the correct gyro angle measurements are augmented with incorrect accelerometer or inclinometer data resulting in the system reporting a level flying aircraft.

Another problem with inclinometers are linear accelerations. In the worst case they are perpendicular to the acceleration vector of gravity. This will result in incorrect angle information.

Inertial Measurement Unit (IMU)

The IMU is, depending on the manufacturer, either a simple sensor unit without angle transformation, or a sensor unit with angle transformation, but no augmentation.
This system has its advantages, if the measuring time is short (drifts) and high system dynamic is expected (augmentation errors).

Attitude and Heading Reference System (AHRS)

The AHRS calculates orientation from gyro data and augments these with accelerometers (inclinometer) data.

Augmentation of the heading is normally done by magnetic compass or GPS.
Because the orientation is augmented, this system has no restrictions on measuring time, but the operating system dynamic is an importing aspect.

Before deciding on an acceptable solution, typical dynamics should be measured and depending on the results, the correct augmentation sensor chosen. Furthermore, typical sensor measurements are indispensable for choosing the correct control algorithm setup.

Inertial Navigation System (INS)

The INS calculates orientation and position from gyro and accelerometer data. INS is often integrated with GPS or DGPS (Differential GPS - more accurate version of GPS).




Much of this paper describes the errors inherent in all sensors and gyros, together with some of the methods available to compensate for them. An appreciation of the nature of these imperfections is of value when the time comes to choose a system for a particular application, and is able to reduce those errors to known and acceptable limits.

This document has been prepared to assist those making decisions about Inertial Sensors and Systems; should there be queries, or suggestions for improving this document, please contact:

GeneSys Elektronik GmbH
In der Spck 10
77656 Offenburg

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