# Inertial guidance system

An inertial navigation system measures the position and altitude of a vehicle by measuring the accelerations and rotations applied to the system's inertial frame. It is widely used because it refers to no real-world item beyond itself. It is therefore immune to jamming and deception. (See relativity and Mach's principle for some background in the physics involved).

An inertial guidance system consists of an inertial navigation system combined with control mechanisms, allowing the path of a vehicle to be controlled according to the position determined by the inertial navigation system. These systems are also referred to as an inertial platform.

 Contents

## Overview

Inertial guidance systems were originally developed for navigating rockets. American rocket pioneer Robert Goddard experimented with rudimentary gyroscopic systems. Dr. Goddard's systems were of great interest to contemporary German pioneers including Wernher von Braun.

A typical inertial navigation system uses a combination of accelerometers, and solves a large set of differential equations to convert these readings into estimates of position and attitude, starting off from a known initial position.

All inertial navigation systems suffer from integration drift, as small errors in measurement are integrated into progressively larger errors in velocity and especially position. This is a problem that is inherent in every open loop control system.

Inertial navigation may also be used to supplement other navigation systems, providing a higher degree of accuracy than is possible with the use of any single navigation system. E.g., if, in terrestrial use, the inertially tracked velocity is intermittently updated to zero by stopping, the position will remain precise for a much longer time, a so-called zero velocity update.

Control theory in general and Kalman filtering in particular, provide a theoretical framework for engineering the fusion of the information from various sensors into an overall predictive model for an inertial navigation system.

## Inertial navigation systems in detail

INSs have angular and linear accelerometers (for changes in position); some include a gyroscopic element (for maintaining an absolute positional reference).

Angular accelerometers measure how the vehicle is twisting in space. Generally, there's at least one sensor for each of the three axes: pitch (nose up and down), yaw (nose left and right) and roll (clockwise or counterclockwise from the cockpit).

Linear accelerometers measure how the vehicle moves. Since it can move in three axes (up & down, left & right, forward & back), it has a linear accelerometer for each axis.

A computer continually calculates the vehicle's current position. First, for each of six axes, it adds the amount of acceleration over the time to figure the current velocity of each of the six axes. Then it adds the distance moved in each of the six axes to figure the current position.

Inertial guidance is impossible without computers. The desire to use inertial guidance in the Minuteman missile and Project Apollo drove early attempts to miniaturize computers.

Inertial guidance systems are now usually combined with satellite navigation systems through a digital filtering system. The inertial system provides short term data, while the satellite system corrects accumulated errors of the inertial system.

An inertial guidance system that will operate near the surface of the earth must incorporate Schuler tuning so that its platform will continue pointing towards the center of the earth as a vehicle moves from place to place.

## Basic schemes

### Gimbaled Gyrostabilized platforms

Some systems place the linear accelerometers on a gimballed gyrostabilized platform. The gimbals are a set of three rings, each with a pair of bearings initially at right angles. They let the platform twist in any rotational axis. There are two gyroscopes (usually) on the platform.

Two gyroscopes are used to cancel gyroscopic precession, the tendency of a gyroscope to twist at right angles to an input force. By mounting a pair of gyroscopes (of the same rotational inertia and spinning at the same speed) at right angles the precessions are cancelled, and the platform will resist twisting.

This system allows a vehicle's roll, pitch and yaw angles to be measured directly at the bearings of the gimbals. Relatively simple electronic circuits can be used to add up the linear accelerations, because the directions of the linear accelerometers do not change.

The big disadvantage of this scheme is that it uses a lot of precision mechanical parts that are expensive. It also has moving parts that can wear out or jam, and is vulnerable to gimbal lock. The primary guidance system of the Apollo spacecraft used a 3-axis gyrostabilized platform, feeding data to the Apollo Guidance Computer. Maneuvers had to be carefully planned to avoid gimbal lock.

### Fluidically Suspended Gyrostabilized Platforms

Gimbal lock constrains maneuvering, and it would be nice to eliminate the slip rings and bearings of the gimbals. Therefore, some systems use fluid bearings or a flotation chamber to mount a gyrostabilized platform. These systems can have very high precisions. Like all gyrostabilized platforms, this system runs well with relatively slow, low-power computers.

The fluid bearings are pads with holes through which pressurized inert gas (such as Helium) or oil press against the spherical shell of the platform. The fluid bearings are very slippery, and the spherical platform can turn freely. There are usually four bearing pads, mounted in a tetrahedral arrangement to support the platform.

In premium systems, the angular sensors are usually specialized transformer coils made in a strip on a flexible printed circuit board. Several coil strips are mounted on great circles around the spherical shell of the gyrostabilized platform. Electronics outside the platform uses similar strip-shaped transformers to read the varying magnetic fields produced by the transformers wrapped around the spherical platform. Whenever a magnetic field changes shape, or moves, it will cut the wires of the coils on the external transformer strips. The cutting generates an electric current in the external strip-shaped coils, and electronics can measure that current to derive angles.

Cheap systems sometimes use bar codes to sense orientations, and use solar cells or a single transformer to power the platform. Some small missiles have powered the platform with light from a window or optic fibers to the motor. A research topic is to suspend the platform with pressure from exhaust gases. Data is returned to the outside world via the transformers, or sometimes LEDs communicating with external photodiodes.

### Strapdown systems

Lightweight digital computers permit the system to eliminate the gimbals, creating "strapdown" systems, so-called because their sensors are simply strapped to the vehicle. This reduces the cost and increases the reliability by eliminating some of the moving parts. Angular accelerometers called "rate gyros" measure how the angular velocity of the vehicle changes. The trigonometry involved is too complex to be accurately performed except by digital electronics. However, digital computers are now so cheap that rate gyro systems are now practical. The Apollo lunar module used a strapdown system in its backup Abort Guidance System (AGS).

## Types of sensors

### Laser gyros

Laser gyroscopes were supposed to eliminate the bearings in the gyroscopes, and thus the last bastion of precision machining and moving parts.

A laser gyro moves laser light in the two opposite directions around a circular path. As the vehicle twists, the light has a doppler effect. The different frequencies of light are mixed, and the difference frequency (the beat frequency) is extractable as an electromagnetic signal whose frequency is proportional to the speed of rotation (Sagnac effect).

In practice, the electromagnetic peaks and valleys of the light lock together. The result is that there is no difference of frequencies, and therefore no measurement.

To unlock the counter-rotating light beams, laser gyros either have independent light paths for the two direction (usually in fiber optic gyros), or the laser gyro is mounted on a sort of audio speaker that rapidly shakes the gyro back and forth to decouple the light waves.

Alas, the shaker is the most accurate, because both light beams use exactly the same path. Thus laser gyros retain moving parts, but they do not move as much.

### Vibrating gyros

Less expensive navigation systems intended for use in automobiles, may use a Vibrating structure gyroscope to detect changes in heading, and the odometer pickup to measure distance covered along the vehicle's track. This type of system is much less accurate than a higher-end INS, but is adequate for the typical automobile application where GPS is the primary navigation system, and dead-reckoning is only needed to fill gaps in GPS coverage when buildings or terrain block the satellite signals.

### Brandy snifter gyros

If a standing wave is induced in a globular brandy snifter, and then the snifter is tilted, the waves continue in the same plane of movement. They don't tilt with the snifter. This trick is used to measure angles. Instead of brandy snifters, the system uses hollow globes machined from piezoelectric materials such as quartz. The electrodes to start and sense the waves are evaporated directly onto the quartz.

This system has almost no moving parts, and is very accurate. However it is still relatively expensive due to the cost of the precision ground and polished hollow quartz spheres.

### Quartz rate sensors

This system is usually integrated on a silicon chip. It has two mass-balanced quartz tuning forks, arranged "handle-to-handle" so forces cancel. Electrodes of aluminum evaporated on the forks and the underlying chip both drive and sense the motion. The system is both manufacturable and inexpensive. Since quartz is dimensionally stable, the system has a good possibility of accuracy.

As the forks are twisted about the axis of the handle, the vibration of the tines tends to continue in the same plane of motion. This motion has to be resisted by electrostatic forces from the electrodes under the tines. By measuring the difference in capacitance between the two tines of a fork, the system can determine the rate of angular motion.

### Pendular accelerometers

The basic accelerometer is just a mass on a spring with a ruler attached. The ruler may be an exotic electromagnetic sensor, but it still senses distance. When the vehicle accelerates, the mass moves, and ruler measures the movement. The bad thing about this scheme is that it needs calibrated springs, and springs are nearly impossible to make consistent.

A trickier system is to measure the force needed to keep the mass from moving. In this scheme, there's still a ruler, but whenever the mass moves, an electric coil pulls on the mass, cancelling the motion. The stronger the pull, the more acceleration there is. The bad thing about this is that very high accelerations, say from explosions, impacts or gunfire, can exceed the capacity of the electronics to cancel. The sensor then loses track of where the vehicle is.

Both sorts of accelerometers have been manufactured as integrated micromachinery on silicon chips.

### Accelerometer-only systems

Some systems use four pendular accelerometers to measure all the possible movements and rotations. Usually, these are mounted with the weights in the corners of a tetrahedron. Thus, these are called "tetrahedral inertial platforms", or TIPs.

When the vehicle rolls, the masses on opposite sides will be accelerated in opposite directions. When the vehicle has linear acceleration, the masses are accelerated in the same direction. The computer keeps track.

TIPs are cheap, lightweight and small, especially when they use micromachined integrated accelerometers. However, as of 2002 they are not very accurate. When they are used, they are often used in small missiles.

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