FAQs
Inertial Navigation Systems are complex to design. Care has to be taken at every stage of the design process. Small errors in the analogue circuitry can lead to large drifts. The signal processing and algorithm programming have to be controlled very precisely.
Inertial Navigation techniques started during the Second World War. The first Inertial Systems used mechanical gyroscopes. These early systems required complicated construction and calibration to precisely balance the gyroscopes. They were prone to failure. Modern spinning gyros used in Inertial Navigation Systems are much more reliable, but even higher reliabilities are being achieved using MEMS sensors.
The original gyroscope systems were platform systems. The gyroscopes held the accelerometers in gimbals. The accelerometers always pointed north, east and down. In modern inertial navigation systems the gyros, or angular rate sensors, and the accelerometers all rotate with the vehicle. This new type of inertial navigation system is called a strapdown inertial navigation system. Strapdown systems can use expensive ring laser gyros, medium-cost fibre optic gyros or low-cost MEMS (Micro Electrical Mechanical System) angular rate sensors. Spinning gyros, such as dynamically tuned gyros, are also usually run in strapdown configurations.
Most Inertial Navigation Systems are designed for military applications, such as fighter jets, cruise missiles, submarines, or tank navigation. Military inertial navigation systems tend to be very expensive when used in civilian applications. They are very hard to export across national boundaries. They are designed for specific purposes and often will not work outside their target application.
More and more civilian applications for inertial navigation systems are appearing, partly as the cost of civilian systems, like the RT3000, reduces. GPS has made it possible for the lower-cost inertial navigation systems to be highly effective. Adding Inertial Navigation to applications where GPS is not reliable enough is a quick way to guarantee reliable position and velocity outputs. Inertial Navigation Systems also make exceptionally accurate measurements of heading, pitch and roll, which are required in many different applications. The acceleration measurements and angular rate measurements needed for inertial navigation are much higher than those required for other applications too.
Conceptual Overview
Any body’s spatial behaviour and motion can be described using three accelerations and three angular rates. Inertial sensors are used to measure the accelerations and the angular rates. Normally the inertial sensors are mounted at 90 degrees to each other in a rigid block. To navigate the angular rate sensors are integrated to give the heading, roll and pitch of the accelerometers. Then the body accelerations are rotated to give north, east and down accelerations. The north, east and down accelerations are integrated to give velocity and the velocity is integrated to give position.
In the diagram the local angular rates are integrated to track the heading, pitch and roll angles. Then the accelerations are rotated to north, east and down accelerations. Finally these accelerations are integrated to give North, East and Down velocities; these velocities are integrated to give Latitude Longitude and altitude.
For navigation on a body like the earth it is essential to have information on the local gravity, the rotation of the body/earth and the curvature of the earth. All these are needed to navigate correctly.
Gravity is very important, since it is such a large acceleration. Gravity has to be removed from the acceleration measurements before integration to velocity. For an inertial navigation system that is aided (or corrected) by GPS, gravity provides a big benefit. Gravity allows the inertial navigation system to measure pitch and roll very accurately using relatively low cost angular rate sensors. This is why low cost, GPS-aided inertial navigation systems tend to have pitch and roll accuracies comparable to high cost, non-aided inertial navigation systems. The same angular rate sensors in a vertical reference unit do not give accurate pitch and roll.
For inertial navigation systems the representation of the rotation from the body-frame (e.g. XYZ accelerations of the vehicle) to the navigation-frame (e.g. North, East, Down accelerations) can be important in some applications. Euler Angles (the angles that would be the same as gimbals) can be used, but these have a problem when the Pitch angle is 90 degrees; when Pitch is 90 degrees the heading bearing and the roll bearing point the same way, so they can rotate freely and are not clearly defined. To solve this the direction-cosine matrix or quaternions can be used. The RT3000 uses Quaternions.
Another problem exists at the North Pole; here every direction is south. Similarly, at the South Pole every direction is north. Many inertial navigation systems that are required to fly over the North or South Pole use a wander angle to solve this problem. The heading may not be valid at the Poles, but the wander angle can be used to compute the Heading correctly when the vehicle leaves.
Mixing all the terms mathematically to give the right results is complicated. Many of the terms appear very small, but small errors give rise to large drift rates in inertial navigation systems. To get the best results it is best to include all the error terms and physical terms in your mathematics.
Analysis of the Errors in Inertial Navigation Systems
Most people’s expectations of Inertial Navigation Systems are that they can remain accurate to metre level for periods of an hour or more. Reality is very different. Here are a few of the errors in an Inertial Navigation System and formulae to compute how the position drifts with time.
In the equations we will use:
se for the position error
ve for the velocity error
ae for the acceleration error
Qe for the orientation error
Velocity Errors
The error growth in an inertial navigation system’s position due to a velocity error is given by the equation:
se(t) = ve(t) * t
For example, if the velocity error is a constant 3 cm/s (just over 0.1 km/h) then the inertial navigation system’s position error will grow by 30cm in 10 seconds, or 5.4 m in three minutes.
Even very accurate inertial navigation systems find it difficult to guarantee a velocity measurement that is better than 0.1 km/h. Yet this error term alone makes the inertial navigation system a lot less accurate than the layman’s expectations. This error growth is independent of the quality of accelerometers or angular rate sensors that are used. It is totally dependent on the quality of the sensors that are used to correct the inertial navigation system. GPS, typically, provides about 0.1 km/h corrections to inertial navigation systems.
Acceleration Errors
The error growth in an inertial navigation system’s position due to an acceleration error is given by the equation:
se(t) = 0.5 * ae(t) * t²
For example, if the acceleration error is a constant 1 mm/s2 (that’s one ten-thousandth of gravity, i.e. not big) then the inertial navigation system’s position error will grow by 5cm in 10 seconds, or 16.2 m in three minutes.
To put this acceleration error in to perspective, an accelerometer with a linearity error of 1% (typical for low-cost accelerometers) will have an acceleration error of about 1% of gravity, which is 100 mm/s2 or 100 times larger than this. This is why low-cost accelerometers are not normally used in inertial navigation systems.
Orientation Errors
The error growth in an inertial navigation system typically couples with gravity and causes an acceleration error. The heading error combines with the speed (or distance travelled) to give position errors too. The acceleration error due to an orientation error (in pitch and roll) is:
ae = g * sin(Qe)
So, it the pitch and roll have a combined error of 0.05 degrees then the acceleration error will be about 9 mm/s2. This error is then applied through the acceleration error formula above. For an orientation error of 0.05 degrees the error growth of an inertial navigation system over 10 seconds is about 45cm and the error growth over three minutes is about 140m.
Bear in mind that, for a GPS-Aided Inertial Navigation System like the RT3000, the position growth should be smaller than this. This is because the accelerometer bias and the roll/pitch accuracy are coupled. This coupling tends to keep the drift smaller since accelerometer bias drift and roll/pitch drift tend to act in opposite directions.
The heading error also has some effect on error growth for a vehicle that is travelling at speed.
se(t) = ve(t) * t * sin(Qe)
For an inertial navigation system with a heading accuracy of 0.1 degrees, travelling at 30 m/s (108 km/h or 67 mph) the error growth is 52cm after 10 seconds or 9.4 metres after three minutes.
It is important to note that this error does not apply in many circumstances. This is because the heading error and the velocity error are usually coupled. If the vehicle has a heading error, but this heading error is coupled to a velocity error in the forward/lateral direction of the vehicle then the error will not apply. The heading error decouples with the velocity error if the vehicle accelerates (for example, turns a 90 degree corner).
Correlated errors apply to other measurements too. For example, if the inertial navigation system has an acceleration error of 10 mm/s2 and cannot separate this error from an orientation error (of about 0.05 degrees) then the drift will be much smaller than 10 mm/s2. This is because the body acceleration has the 10 mm/s2 error, but the acceleration in the north and east directions can be more accurately known.
There are other errors that apply to inertial navigation systems too, but these have not been dealt with here.
