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This report serves as a brief overview of Inertial navigation Systems (INS) in respect of aircraft industry. Modern INS outdated all other navigation systems so far. These consist of a set of gyros and accelerometers which measure the aircraft's angular and linear motion and work with a computing system which computes aircraft's heading and attitude from the gyro and accelerometer outputs, given that, initial position and velocity of the aircraft are provided from another source. Different types of gyroscopes and accelerometers, followed by their mechanism, errors and the ways to overcome those errors are explained in this report.
Inertial navigation is the process of establishing the position, velocity, heading and attitude of a vehicle using information derived from internal sensors. The operation of inertial sensors depends upon the laws of classical mechanics as formulated by Sir Isaac Newton which states that the motion of a body will continue uniformly in a straight line until disturbed by an external force acting on the body. The law also tells us that this force will produce a proportional acceleration of the body. Inertial measurement units(IMU) usually contains three orthogonal rate- gyroscopes and accelerometers measuring angular velocity and linear acceleration respectively relative to a known starting point, velocity and orientation using Newton's law.
Hence , Inertial navigation is the process whereby the measurements provided by gyroscopes and accelerometers are used to determine the position of the vehicle in which they are installed. By combining the two sets of measurements, it is possible to define the translational motion of the vehicle within the inertial reference frame and so to calculate its position within it.
INS was first used on rockets in the 1940's. In 1996, inertial navigation systems were widely used in military vehicles. Many ships, submarines, guided missiles, space vehicles and all modern military are equipped with INS due to its immunity.
Inertial navigation system arrangement
INS uses two types of configuration. The only difference between them is the frame in which the sensors operate. Both of them are described below.
Inertial navigation technology originally used stable platform techniques. In this configuration, inertial sensors are mounted on a platform. The platform is isolated from the rotational motion of the vehicle using a number of gimbals arranged to provide at least three degrees of rotational freedom. The movement of these gimbals is controlled by torque motors. Those motors are activated by information provided by gyroscopes as it detects any platform rotation. Thus, the platform is kept aligned with the global frame.
Figure 1: Authors illustration of block diagram of stabilised platform
In this system, the inertial sensors are strapped directly on the aircraft body and are not isolated from its angular motion. Thus, gimballed platform is not required for this system. But, it uses a computer to establish and resolve the inertial data which reduces the mechanical complexity of the system.
A gyroscope is a device which acts as a rotating body and thus measure or maintains orientation, based on the principles of conservation of angular momentum. It is used in various applications to sense either the angle turned through by an aircraft or more commonly, its angular rate of turn about some defined axis. A modern gyroscope can fulfill each of the tasks stated below:
Flight path sensor or platform stabilization
Figure 2: Authors illustration of gimballed gyroscope.
There are several phenomena on which the operation of gyroscope depends but it usually exhibits three fundamental properties, namely gyroscopic inertia, angular momentum and precession.
Gyroscopic inertia is fundamental to the operation of all spinning mass gyroscopes, as it defines a direction in space that remains fixed in the inertial reference frame, that is, fixed in relation to a system of coordinates which do not accelerate with respect to the 'fixed stars'. The establishment of a fixed direction enables rotation to be detected, by making reference to this fixed direction. The rotation of an inertial element generates an angular momentum vector which remains fixed in space, given perfection in the construction of gyroscope.
Angular momentum is defined by the distribution of mass on a rotor as well as by its angular velocity. The angular momentum (H) of a rotating body is the product of its moment of inertia (I) and its angular velocity (Ï‰), that is,
H = IÏ‰
Where I is the sum of the products of the mass elements that make up the rotor and the square of their distances from the given axis.
Precession is the rotation of the gimbals, relative to inertial space. This rotation is produced jointly by the angular momentum of the rotating body and the applied force. In the case of a freely spinning body, such as the Earth (or the rotation of an electrostatic gyroscope), there is not a material frame with spin bearings. In this case, the precession must be considered to be that of the axis system which an imaginary gimbal would have - one axis through the north and south poles, and two mutually orthogonal in the plane of the Equator.
A mechanical gyroscope calculates orientation based on the principle of conservation of the angular momentum. The disc is mounted on a frame to minimize the external moments (i.e. due to friction). This allows the target to turn around the disc without causing any change in the direction of its axis. The orientation of the target then can be computed from the angles shown by rotational encoders mounted on the frame. Each gyroscope gives us one reference axis in space. At least two gyroscopes are needed to find the orientation of an object in space.
Figure 3: Authors illustration of mechanical gyroscope.
Advantage & disadvantages of mechanical gyroscopes:
Main advantage of this tracking system is that it does not require any external reference to work. Because the axis of the rotating wheel acts as the reference. The drawback of this system is its configuration. Because of the moving parts causing friction, the inertial momentum of the wheel does not remain parallel to the axis of rotation. This causes a drift in the direction of the wheel axis with time. Taking relative measurements of the orientation rather than absolute measurements can minimize this drift. As a consequence, the system suffers from accumulated numerical errors but a periodic re-calibration of the system will insure, more accuracy over time. Lubricants are used to minimize the friction which increase the cost of the device.
Solid state gyroscopes
The term 'Solid state' stands for an electronic device in which the flow of electrical current is through solid material and not through a vacuum. So solid state gyroscopes use flow of electric current through solid material to measure orientation of the attached object.
Discovered in 1913, the Sagnac effect found its first practical application several decades ago in the ring laser gyroscope (RLG), now used extensively in commercial inertial navigation systems for aircraft. But, since this implementation requires high vacuum and
precision mirror technology, cost has been a factor limiting its application. 'Sagnac effect' plays a vital role in solid state gyroscopes which is named after the French physicist G.Sagnac. This states that the resulting difference in the transit times for laser light waves travelling around a closed path in opposite direction is proportional to the input rotation rate.
Nowadays, lots of solid state gyroscopes are being used in the industry. Mostly used gyroscopes are described below:
Fibre optic gyroscopes(FOG)
Fibre optic gyroscopes sense angular motion using interference of light. Such devices often use the visible wavelengths, but it can also operate in the near infrared. It is dependent on the formation of a Sagnac interferometer  In its simplest form, light from a broad band source is split into two beams that propagate in opposite directions around an optical fibre coil. These two beams are then combined at a second beam splitter to form an interference pattern where the resultant intensity is observed using a photo-detector. The phase
Figure 4: Authors illustration of FOG.
shift introduced due to the Sagnac effect. They are combined when the beams exit the fibre. The resulting phase difference results in a change in amplitude of the interference pattern formed when the two beams are recombined.
184.108.40.206. Errors and errors reduction
A bias or drift occurs due to changes in ambient temperature which cause a multitude of effects within the sensor. To minimize this error, the expansion coefficient of the fibre and the coil former should be well matched otherwise differential stress will be induced by thermal expansion which will result in measurement error.
The presence of any stray magnetic fields can have several adverse effects on the gyroscopes like interaction with non-optical components causing Faraday effect which changes the state of polarization of the light in optical fibre. Use of magnetic shielding can minimise this problem.
Ring laser gyroscopes
A ring laser gyroscope wherein a first and a second laser beam propagate with propagating directions different with each other comprises electrode areas on an optical waveguide configuring the ring laser and controls an current injected or a voltage applied to the electrode areas, wherein the oscillating frequencies of the first and second laser beams are different from each other, thereby causing an increase and a decrease in the beat frequency enabling to detect the direction and the speed of a rotation at the same time. With regards to a method for detecting a rotation, the anode of the laser gyro is connected to an operational amplifier. Since the signal outputted from the operational amplifier has a frequency corresponding to the angular speed, it is converted into the voltage by a frequency-voltage conversion circuit so as to detect a rotation.
Errors and error reduction:
The 'Lock-in' problem should be overcome by the RLG which arises due to imperfection in the lasing cavity, mainly in mirrors. It causes scale factor error which tends to pull the frequencies of the two beams together at low rotation rates. If the input rate in the RLG drops below a threshold is known as 'Lock-in rate'. The two beams lock together at the same frequency resulting zero output and a dead zone. This lock-in dead zone is of the order of 0.01 to 0.1 /s compared with 0.01 /hr accuracy required for an INS. A very effective method of overcoming this problem is to mechanically dither the laser block about the input axis at a typically frequency about 100 Hz with a peak velocity of about 100 /s (corresponding to amplitude of 1.5 arc second approximately)
Micro machined silicon gyroscopes(MEMS)
MEMS gyroscopes are introduced in the modern navigation system due to their low production cost and very simple configuration. It is build on Coriolis effect stating that a object of mass m rotating at angular velocity Ï‰ moving with velocity v experiences a force,
F= 2m(Ï‰ x v)
It contains vibrating elements to measure this effect. A secondary vibration is induced along the perpendicular axis, when the gyroscope is rotated. The angular velocity is calculated by measuring this rotation.
Errors and error reduction
The major disadvantage of MEMS gyroscopes is that they are very less accurate than optical devices. As technology improving, this gyroscope are becoming more and more accurate and reliable.
As described before, INS relies upon the measurement of acceleration which can be determined by accelerometer. An accelerometer works on Newton's second law of motion. A force F acting on a body of mass m causes the body to accelerate with respect to inertial space. This acceleration (a) is given by,
F = ma = mf + mg
Where f is the acceleration produced by forces other than gravitational field.
Mechanical accelerometers are mainly mass - spring type devices. INS is using these sensors for long time. Different construction techniques have been implied to use in different environments.
Figure 5: Authors illustration of accelerometer.
Mechanical accelerometers can be operated in two different types of configuration: either open or closed loop configuration.
Open loop configuration
A proof mass is suspended in a case and confined to a zero position by means of a spring. Additionally, damping is applied to give this mass and spring system a realistic response corresponding to a proper dynamic transfer function. When the accelerations are applied to the case of the sensor, the proof mass is deflected with respect to its zero or 'null' position and the resultant spring force provides the necessary acceleration of the proof mass to move it with the case. For a single - axis sensor, the displacement of the proof mass with respect to its 'null' position within the case is proportional to the specific force applied along its input. A more accurate version of this type of sensor is obtained by nulling the displacement of the pendulum., since 'null' position can be measured more accurately than displacements.
Closed loop accelerometer
The spring is replaced by an electromagnetic device that produces a force on the proof mass to maintain it at its 'null' position. Usually, a pair of coils is mounted on the proof mass within a strong magnetic field. When a deflection is sensed, an electric current is passed through the coils in order to produce a force to return the proof mass to its 'null' position. Magnitude of the current in the coils is proportional to the specific force sensed along the input axis.
All accelerometers are subjected to errors which limit the accuracy of the force being measured. The major sources of error in mechanical errors are listed below:
Fixed bias: this is a bias or displacement from zero on the measurement of specific force which is present when the applied acceleration is zero.
Scale-factor errors: This is the error in the ratio of a change in the output signal to a change in the input acceleration.
Cross-coupling errors: These errors arise as a result of manufacturing imperfection. Erroneous accelerometer outputs resulting from accelerometer sensitivity to accelerations applied normal to the input axis.
Due to those errors of mechanical accelerometers, researchers are giving their best effort to investigate various phenomena to produce a solid-state accelerometer. They came up with various types of devices so far, among those surface acoustic wave, silicon and quartz devices(Vibratory devices) were most successful. Good things about these sensors are that they are small, rugged, reliable and convenient with strapdown applications. These three types of solid-state accelerometers are described below.
Surface acoustic wave(SAW) accelerometer
This is an open-loop instrument which consist of a piezoelectric quartz cantilever beam which is fixed at one end of the case but movable at the other end, where the proof mass is rigidly attached. The beam bends responding to the acceleration applied along the input axis. Due to this, frequency of the SAW is changed. Acceleration can be determined by measuring the change in frequency.
Figure 6: Authors illustration of SAW accelerometer.
Errors and error reduction:
The effects of temperature and other effects of a temporal nature can be minimised by generating the reference frequency from a second oscillator on the same beam.
Lock- in type effects are mainly prevented by ensuring that this reference signal is at a slightly different frequency from that used as the 'sensitive' frequency. 
Single-crystal silicon forms the frame, hinges and proof mass. Anodic bonding joins this piece to metalized wafers which enclose the accelerometer and also serve as electrodes for sensing proof mass motion and for rebalancing. Electrostatic centering of the proof mass obviates the need for magnetic materials and coils. When the accelerometer is rebalanced using voltage forcing, a potential is applied to the pendulum and to one or both electrodes. The voltage establish electric fields that induce charge on the nonconductive pendulum. This causes a net force to act on the proof mass. Thus, the force generated is a function of the square of the applied voltage and of the gap between the pendulum and the electrode.
These are open-loop devices which use quartz crystal technology. They are consist of a pair of quartz crystal beams, each supporting a proof mass pendulum and are mounted symmetrically back-to-back. When an acceleration is applied, one beam is compressed while the other stretched.
Figure 7: Authors illustration of Vibratory accelerometer.
The compressed beam experienced a decrease in frequency while the stretched one experience the opposite. The difference between these two frequency is directly proportional to the acceleration applied.
Errors and error reduction:
Most of the errors of this sensor can be minimized by designing carefully. Instead of using one beam, several symmetrically arranged beams can reduce errors.
According to the data collected within this report, it is clear to see the INS system has helped a lot towards the modernization of navigation system. Further improvement in MEMS technology can open several doors in aviation systems. Its high accuracy and self contained rate made it immune to any obstacle.
Inertial navigation system has improved a lot in past 5 decades. It has helped the aviation Industry to improve navigation system and thus ease the duty of pilots.