Dead reckoning

Dead reckoning (DR) is the process of estimating one's current position based upon a previously determined position, or fix, and advancing that position based upon known or estimated speeds over elapsed time, and course. While traditional methods of dead reckoning are no longer considered primary means of navigation, modern inertial navigation systems, which also depend upon dead reckoning, are very widely used.

A disadvantage of dead reckoning is that since new positions are calculated solely from previous positions, the errors of the process are cumulative, so the error in the position fix grows with time.

Animal navigation
In studies of animal navigation, dead reckoning is more commonly (though not exclusively) known as path integration, and animals use it to estimate their current location based on the movements they made since their last known location. Animals such as ants, rodents, and geese have also been shown to continuously keep track of their locations relative to a starting point and return to it, an important skill to have for creatures that forage for food and then return to a fixed home.

Marine navigation
In marine navigation a "dead" reckoning plot generally does not take into account the effect of currents or wind. Aboard ship a dead reckoning plot is considered important in evaluating position information and planning the movement of the vessel. Dead reckoning begins with a known position, or fix, which is then advanced, mathematically or directly on the chart, by means of recorded heading, speed, and time. Speed can be determined by many methods. Before modern instrumentation, it was determined aboard ship using a chip log. More modern methods include pit log referencing engine speed (e.g. in rpm) against a table of total displacement (for ships) or referencing one's indicated airspeed fed by the pressure from a Pitot tube. This measurement is converted to an equivalent airspeed based upon known atmospheric conditions and measured errors in the indicated airspeed system. A naval vessel uses a device called a pit sword (rodmeter), which uses two sensors on a metal rod to measure the electromagnetic variance caused by the ship moving through water. This change is then converted to ship's speed. Distance is determined by multiplying the speed and the time. This initial position can then be adjusted resulting in an estimated  position by taking into account the current (known as set and drift in marine navigation). If there is no positional information available, a new dead reckoning plot may start from an estimated position. In this case subsequent dead reckoning positions will have taken into account estimated set and drift.

Dead reckoning positions are calculated at predetermined intervals, and are maintained between fixes. The duration of the interval varies. Factors including one's speed made good and the nature of heading and other course changes, and the navigator's judgment determine when dead reckoning positions are calculated.

Before the development of the marine chronometer, dead reckoning was the primary method of determining longitude available to mariners such as Christopher Columbus and John Cabot on their trans-Atlantic voyages. Tools such as the Traverse board were developed to enable even illiterate crew members collect the data needed for dead reckoning.

Air navigation
Traditionally, in air navigation, displacement or position caused by wind were taken into account, using a tool called a wind triangle. Generally speaking, dead reckoning positions were calculated at least once every 300 miles and when making combined turns totaling more than 30 degrees from the initial heading out of the last DR position.

Today, dead reckoning is rarely used in this traditional form for air navigation, but it survives in the form of inertial navigation systems, which are nearly universal on more advanced aircraft. The INS is used in combination with other navigation aids, such as GPS, in order to provide reliable navigation capability under virtually any conditions, with or without external navigation references.

Automotive navigation
Dead reckoning is today implemented in some high-end automotive navigation systems in order to overcome the limitations of GNSS technology alone. Satellite microwave signals are unavailable in parking garages and tunnels, and often severely degraded in urban canyons and near trees due to blocked lines of sight to the satellites or multipath propagation. In a dead-reckoning navigation system, the car is equipped with sensors that record the wheel rotation and steering direction. These sensors are often already present in cars for other purposes (anti-lock braking system, electronic stability control) and can be read by the navigation system from the controller-area network bus. The navigation system then uses a Kalman filter in order to integrate the reliability and short-term accuracy of the sensor data with the long-term accuracy of the satellite data into a position fix that combines the best of both worlds.

Etymology
There is disagreement about the derivation of the phrase. It is sometimes thought (particularly among sailors and pilots in the USA ) to come from deduced reckoning and is sometimes given in modern sources as an alternatively spelled ded reckoning; however, according to the Oxford English Dictionary, the phrase dead reckoning dates from Elizabethan times (1605–1615), and no trace of it can be found in books published before the 20th Century.

The popular etymology from deduced is not documented in the Oxford English Dictionary or any other historical dictionary. It is not clear what 'dead' might mean in the context of dead reckoning, and several alternative theories have been promulgated.

Math
Here are the dead reckoning formulas for the coordinates (x and y), and heading($$\theta$$) for a differential drive robot with encoders on both drives:


 * $$\Delta x = R_W Cos(\theta)(T_1+T_2)\,$$


 * $$\Delta y = R_W Sin(\theta)(T_1+T_2)\,$$


 * $$\Delta \theta = 2 \pi \frac{T_1-T_2} {T_R}$$

where $$T_1$$ are the encoder ticks recorded on drive one, $$T_2$$ are the encoder ticks recorded on drive two, $$R_W$$ is the radius of the drive wheel, and $$T_R$$ is the number of encoder ticks recorded in a full, in-place rotation.