In naval gunnery, when long-range guns became available, an enemy ship would move some distance after the shells were fired. It became necessary to figure out where the enemy ship, the target, was going to be when the shells arrived. The process of keeping track of where the ship was likely to be was called rangekeeping, because the distance to the target—the range—was a very important factor in aiming the guns accurately. As time passed, train (also called bearing), the direction to the target, also became part of rangekeeping, but tradition kept the term alive. Rangekeeping is an excellent example of the application of analog computing to a real-world mathematical modeling problem. Because nations had so much money invested in their capital ships, they were willing to invest enormous amounts of money in the development of rangekeeping hardware to ensure that the guns of these ships could put their projectiles on target. This article presents an overview of the rangekeeping as a mathematical modeling problem. To make this discussion more concrete, the Ford Mk 1 Rangekeeper is used as the focus of this discussion. The Ford Mk 1 Rangekeeper was first deployed on the USS Texas in 1916 during World War I. This is a relatively well documented rangekeeper that had a long service life. While an early form of mechanical rangekeeper, it does illustrate all the basic principles. The rangekeepers of other nations used similar algorithms for computing gun angles, but often differed dramatically in their operational use. In addition to long range gunnery, the launching of torpedoes also requires a rangekeeping-like function. The US Navy during World War II had the TDC, which was the only World War II-era submarine torpedo fire control system to incorporate a mechanical rangekeeper (other navies depended on manual methods). There were also rangekeeping devices for use with surface ship-launched torpedoes. For a view of rangekeeping outside that of the US Navy, there is a detailed reference that discusses the rangekeeping mathematics associated with torpedo fire control in the Imperial Japanese Navy.
The following discussion is patterned after the presentations in World War II US Navy gunnery manuals.
Analysis
Coordinate system
US Navy rangekeepers during World War II used a moving coordinate system based on the line of sight (LOS) between the ship firing its gun (known as the "own ship") and the target (known as the "target"). As is shown in Figure 1, the rangekeeper defines the "y axis" as the LOS and the "x axis" as a perpendicular to the LOS with the origin of the two axes centered on the target. An important aspect of the choice of coordinate system is understanding the signs of the various rates. The rate of bearing change is positive in the clockwise direction. The rate of range is positive for increasing target range.
Target tracking
General approach During World War II, tracking a target meant knowing continuously the target's range and bearing. These target parameters were sampled periodically by sailors manning gun directors and radar systems, who then fed the data into a rangekeeper. The rangekeeper performed a linear extrapolation of the target range and bearing as a function of time based on the target information samples. In addition to ship-board target observations, rangekeepers could also take input from spotting aircraft or even manned balloons tethered to the own ship. These spotting platforms could be launched and recovered from large warships, like battleships. In general, target observations made by shipboard instruments were preferred for targets at ranges of less than 20,000 yards and aircraft observations were preferred for longer range targets. After World War II, helicopters became available and the need to conduct the dangerous operations of launching and recovering spotting aircraft or balloons was eliminated (see Iowa-class battleship for a brief discussion). During World War I, target tracking information was often presented on a sheet of paper. During World War II, the tracking information could be displayed on electronic displays (see Essex-class aircraft carrier for a discussion of the common displays).
Target range Early in World War II, the range to the target was measured by optical rangefinders. Though some night operations were conducted using searchlights and star shells, in general optical rangefinders were limited to daytime operation. During the latter part of World War II, radar was used to determine the range to the target. Radar proved to be more accurate than the optical rangefinders (at least under operational conditions) and was the preferred way to determine target range during both night and day.
Target speed Early in World War II, target range and bearing measurements were taken over a period of time and plotted manually on a chart.
The speed and course of the target could be computed using the distance the target traveled over an interval of time. During the latter part of World War II, the speed of the target could be measured using radar data. Radar provided accurate bearing rate, range, and radial speed, which was converted to target course and speed. In some cases, such as with submarines, the target speed could be estimated using sonar data. For example, the sonar operator could measure the propeller turn rate acoustically and, knowing the ship's class, compute the ship's speed (see TDC for more information).
Target course
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