By established principles of mathematical physics, the velocity of an object is defined as the rate of change of its position, and the acceleration is defined as the rate of change of the velocity. These relations can be applied to the navigational problem of position finding if an instrument can be devised to measure acceleration and then to convert it successively to velocity and to position. In the terminology of calculus, acceleration is integrated (summed a little at a time) to get velocity, then velocity is integrated to get position.
In one form of accelerometer, a reference mass is suspended on springs within a housing firmly attached to the craft. The inertia of the mass causes it to tend to remain stationary, but any acceleration of the craft tends to displace the housing relative to the mass. The forces required to nullify relative motion of the mass and the housing—in three directions fixed by gyroscopes—can be measured electrically. The electrical signals are directly related to the forces and, by Newton’s second law of motion, to the accelerations. Standard electronic circuitry performs the necessary integrations of the accelerations to provide the distances and directions, in three dimensions, through which the craft has moved from its original position.
Such combinations of accelerometers coupled with integrators are called inertial guidance systems; in the context of navigation, they amount to sophisticated dead-reckoning devices. Since their introduction, starting in 1950, they have proved extremely valuable in controlling trajectories of submarines, booster rockets, and spacecraft. Their errors, like those of any other dead-reckoning system, are cumulative with time, but nuclear-powered submarines have traveled under the north polar ice cap, guided solely by inertial systems, with errors of less than a mile per week.
Dead reckoning by computer
In modern inertial navigation systems, computers have proved well-suited to processing the streams of data—directions, speeds, and times—involved in keeping track of position. In military land vehicles, computers are fed by compasses and wheel-mounted sensors. Navigators aboard ships depend on the gyrocompass and the log; those in aircraft rely on the gyromagnetic compass and Doppler-effect speed measurements. The computers can be programmed to display or print periodically updated positional information. Inertial guidance systems may provide dead-reckoning information only, though compass and Doppler data can be combined with inertial outputs. Information from radio navigation systems, such as loran or the global positioning system (GPS), can be added to the dead reckoning.
Radio navigation systems that can provide continuous indication of position are eliminating the distinction between position fixing and dead reckoning. Navigation accuracies are improving by supplying both the classical dead-reckoning data (speed, direction, altitude rates, and angles) and the continuously updated position to a computer, which determines the speed, heading, and rate of climb or descent that must be maintained to execute the flight plan. Many computers apply the technique called Kalman filtering, which weights each of the several supplied data according to its expected quality and uses previous position and velocity solutions in determining the current best estimate of position and other desired quantities.
Originally, analog computers were used in navigation systems, and calculations of a relatively simple nature involving inputs from various electrical sources were continuously performed. Today, digital computers are employed for performing virtually all the necessary calculations. The digital computer works so fast that for navigational purposes it can be considered virtually instantaneous and can therefore provide continuous information for control purposes. It has a memory to store information for use when needed. It is built from electronic modules that are mass-produced at low cost. It has only one disadvantage. Conversion of analog information into digital form can be costly. Hence, although far superior to the analog computer, it is less economical whenever a large number of electrical signals must be combined in relatively simple ways without any need for memory. Such situations still apply to control systems in many craft.
Radio navigation
Direction finders
To avoid any navigation hazards marked on the charts, a mariner needs to know the vessel’s exact position. By means of a sight fitted to the compass, the direction of any visible landmark or buoy can be measured. This direction, called a bearing, can be marked on the chart as a line passing through the identified reference point. A similar line corresponding to a second bearing will intersect the first and fix the position of the vessel relative to the navigation hazard, as shown in the .
The invention of radio transmission and reception led to an improvement in this navigational technique, making it possible to obtain bearings from reference points obscured by fog or darkness. The signals picked up by a loop antenna are weakest when the plane of the loop is perpendicular to the direction in which the radio waves are traveling. If the receiver is tuned to the frequency of a particular transmitter and the loop is rotated for minimum signal pickup, the direction to the transmitter can be found and plotted. When the procedure is repeated with another transmitter, the second bearing will intersect the first, thus fixing the navigator’s position, as before.
Soon after ships were first equipped with radio, direction-finding stations were placed on shore at strategic points along navigational routes and near harbour approaches. Upon receiving a request by radio from a ship, two or more shore stations determined the directions from which the ship’s signal arrived and transmitted this information to the vessel. This allowed the navigator to fix the ship’s position. The limitation of this service to one vessel at a time, however, was a serious drawback in bad weather, when demands were heavy. Beginning in 1921, continuously operating transmitters were placed ashore and the direction finder placed on the ship to eliminate the possibility of overloading the system and to give the navigator two further advantages: that of taking continuous or frequent bearings on any shore beacon and that of taking bearings of any receivable signal, such as transmissions from commercial broadcasting stations and from other vessels. This change in the system was roughly coincident with the initial growth of aviation, and the airborne direction finder immediately became a valuable aid to air navigation.
Under ideal conditions a well-designed direction finder will provide bearings within 1° or 2° of the true value. The uncertainty can be considerably increased, however, if the direction of the radio waves is altered by reflections from the ionosphere or refraction in the atmosphere.
The loop-antenna radio direction finder, almost as old as radio itself, developed into a device in which a motor turned the loop, and electronic circuitry identified the direction of the source of the signals. This instrument, originally called a radio compass, could guide the navigator toward any detectable transmitter. It was often linked to a compass so as to display not merely the direction of the radio station compared to the heading of the craft but the actual direction as plotted on a chart.
Radio-beam systems
The directional selectivity of loop antennas when they are used as receivers is duplicated when they are used as transmitters. Such an antenna can be oriented so that it radiates strong signals to the north and south but practically none to the east or west. Pathway-defining ground stations for aircraft were developed during the 1920s and ’30s. They were equipped with loop antennas in pairs at right angles, arranged so that one antenna broadcast the International Morse Code character A (· —) and the other broadcast the character N (— ·). Midway between the directions in which only A or only N could be heard, the characters interleaved to produce a steady tone; these four intermediate directions were the preferred courses, called beams. Only a slight deviation of the receiver from a beam disrupted the steady tone, and the direction in which the craft was off the beam was indicated by the predominance of one Morse character or the other. The pilot flew in one of the four directions toward or away from the transmitting beacon, which was called a four-course beacon or a radio range.
The distance at which the signals could be detected was limited, and the four-course beacons were replaced by VOR (very-high-frequency omnidirectional range), the beacons of which operated on an entirely different principle. At each beacon, one antenna sent out waves that had the same intensity in all directions. A second antenna rotated and sent out a narrow beam of waves that, when directed north, coincided in phase with those of the first antenna; that is, the peaks of the waves from the two antennas reached the receiver at the same instant. When the rotating beam pointed east, the two sets of waves were out of phase by 90° (one quarter of a wavelength); when the beam pointed south, the phase difference was 180°; and so on. A receiver in the aircraft measured the phase difference and displayed the bearing of the VOR beacon along with the heading of the aircraft.
From the radio range, with its so-called beams, true beam systems have developed. In these the loops are replaced by improved antennas that concentrate the radio waves into narrow beams a few degrees wide; the dots and dashes are replaced by more sophisticated patterns or modulations. In the instrument landing system (ILS), used to help aircraft approach and land on an airfield, the two antennas transmit waves about 10 feet (3 metres) long. These waves, though shorter than those employed in earlier systems, necessitate antenna structures about 100 feet (30 metres) long on the ground. Some such installations make it possible for suitably equipped aircraft to land in conditions of practically zero visibility. The beams point in almost the same direction, and, once the aircraft has entered the beams, an ILS receiver on the airplane can measure the angular displacement from the centre line and display this displacement on an instrument or use it to guide the aircraft along a line toward the point of landing. In addition to the steering beams, which make up the localizer element of the ILS, there are two similar but even narrower beams transmitted in the vertical plane that guide the aircraft down the correct slope toward the point of touchdown.
The microwave landing system (MLS) uses modulated wavelengths that are only about a half inch (one centimetre) long. One beam sweeps side-to-side while the other sweeps up-and-down. Unlike the ILS, the MLS, with its dynamic beam geometry, allows airplanes to follow various descent angles and travel along curved or segmented trajectories. Several microwave landing systems have been installed at commercial airports around the world.
