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The navigator plots his 9am position, indicated by the triangle, and, using his course and speed, estimates his position at 9:30am and 10am

In navigation, dead reckoning (also ded (for deduced) reckoning or DR) is the process of calculating one's current position by using a previously determined position, or fix, and advancing that position based upon known or estimated speeds over elapsed time, and course.

Drift is the angle between the heading of the airplane and the desired track. A is the last known position (fix, usually shown with a circle). B is the air position (usually shown with a plus sign). C is the DR position (usually shown with a triangle).

Dead reckoning is subject to cumulative errors. Advances in navigational aids which give accurate information on position, in particular satellite navigation using the Global Positioning System, have made simple dead reckoning by humans obsolete for most purposes. However, inertial navigation systems, which provide very accurate directional information, use dead reckoning and are very widely applied.

By analogy with their navigational use, the words dead reckoning are also used to mean the process of estimating the value of any variable quantity by using an earlier value and adding whatever changes have occurred in the meantime. Often, this usage implies that the changes are not known accurately. The earlier value and the changes may be measured or calculated quantities.

There is speculation on the etymological origin of the term, but no reliable information.

## Errors

Dead reckoning may give the best available information on position, but is subject to significant errors due to many factors as both speed and direction must be accurately known at all instants for position to be determined accurately. For example, if displacement is measured by the number of rotations of a wheel, any discrepancy between the actual and assumed diameter, due perhaps to the degree of inflation and wear, will be a source of error. As each estimate of position is relative to the previous one, errors are cumulative.

In studies of animal navigation, dead reckoning is more commonly (though not exclusively) known as path integration. Animals use it to estimate their current location based on their movements from their last known location. Animals such as ants, rodents, and geese have been shown to track their locations continuously relative to a starting point and to return to it, an important skill for foragers with a fixed home.[1][2]

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.[3]

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 19th-century development of the marine chronometer and the lunar distance method, 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 to collect the data needed for dead reckoning. Polynesian navigation, however, uses different wayfinding techniques.

a. Ground position
b. Air position
c. DR position

Before the advent of modern aids dead reckoning was widely used in air navigation, taking into account displacement of position caused by wind as far as possible, often using a tool called a wind triangle. As a rule dead reckoning (DR) 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.

Simple dead reckoning fell out of use for air navigation, but is used by inertial navigation systems (INSes), 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, without the need for external navigation references.

However, simple dead reckoning is still widely used by civil aircraft not equipped with GPS or radio navigation aids. The pilot periodically establishes a location fix from visual sighting of landmarks with reference to a map, allowing errors in dead reckoned course to be corrected. Pilots of advanced aircraft are trained in dead reckoning, which remains usable in case of failure of advanced navigational systems.

Ded (as in deduced) reckoning is still on the curriculum for VFR (visual flight rules - or basic level) pilots in Australia. It was experienced during a navigation exercise with Peter Binis Advanced Flight Training with master instructor Steve Pearce in January 2013. He encouraged its use, for diversions. The argument is that once you have a fix (on your position) a knowledge of time/distance and an approximate heading will serve well.

A moderately experienced pilot can estimate the heading and distance without rulers and protractors. Look at the chart, estimate the angle of change, and roughly estimate the distance with a thumb or finger - e.g. A thumb-width might be five nautical miles. Divide the distance by two for an aircraft flying 120kts, and you have the number of minutes to fly that heading. Though not perfect, it can be remarkably reliable.

There is also an argument that the entire method of navigating by charts, adjusted for wind and expected speed, is in fact ded reckoning. This 'analogue' method, without the aid of GPS and/or instrument flight rules paths, is at best an estimate in need of constant adjustment. Again, this is very much current teaching practice for VFR pilots in Australia and elsewhere.

It is taught regardless of whether the aircraft has navigation aids such as GPS, ADF and VOR.

Dead reckoning is today implemented in some high-end automotive navigation systems in order to overcome the limitations of GPS/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 know the wheel diameter and record wheel rotations 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 to integrate the always-available sensor data with the accurate but occasionally unavailable position information from the satellite data into a combined position fix.

## Autonomous navigation in robotics

Dead reckoning is utilized in some lower-end, non mission-critical, or tightly constrained by time or weight, robotic applications. It is usually used to reduce the need for sensing technology, such as ultrasonic sensors, GPS, or placement of some linear and rotary encoders, in an autonomous robot, thus greatly reducing cost and complexity at the expense of performance and repeatability. The proper utilization of dead reckoning in this sense would be to supply a known percentage of electrical power or hydraulic pressure to the robot's drive motors over a given amount of time from a general starting point. Dead reckoning is not totally accurate, which can lead to errors in distance estimates ranging from a few millimeters (in CNC machining) to kilometers (in UAV's), based upon the duration of the run, the speed of the robot, the length of the run, and several other factors.

## Directional dead reckoning

The south-pointing chariot was an ancient Chinese device consisting of a two-wheeled horse-drawn vehicle which carried a pointer that was intended always to aim to the south, no matter how the chariot turned. The chariot predated the navigational use of the magnetic compass, and could not detect the direction that was south. Instead it used a kind of directional dead reckoning: at the start of a journey, the pointer was aimed southward by hand, using local knowledge or astronomical observations e.g. of the Pole Star. Then, as it travelled, a mechanism possibly containing differential gears used the different rotational speeds of the two wheels to turn the pointer relative to the body of the chariot by the angle of turns made (subject to available mechanical accuracy), keeping the pointer aiming in its original direction, to the south. Errors, as always with dead reckoning, would accumulate as distance travelled increased.

## Differential steer drive dead reckoning

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

$\Delta \theta = 2 \pi \frac{R_W} {D} \frac{T_1-T_2} {T_R}$
$\Delta x = R_W \cos(\theta)(T_1+T_2) \frac{\pi} {T_R}$
$\Delta y = R_W \sin(\theta)(T_1+T_2) \frac{\pi} {T_R}$

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 each drive wheel, $D$ is the separation between the wheels, and $T_R$ is the number of encoder ticks recorded in a full rotation of a wheel.

## Dead reckoning for networked games

Networked games and simulation tools routinely use dead reckoning to predict where an actor should be right now, using its last known kinematic state (position, velocity, acceleration, orientation, and angular velocity).[4] This is primarily needed because it is impractical to send network updates at the rate that most games run, 60 Hz. The basic solution starts by projecting into the future using linear physics:[5]

$P_t = P_0 + V_0T + \frac{1}{2}A_0T^2$

This formula is used to move the object until a new update is received over the network. At that point, the problem is that there are now two kinematic states: the currently estimated position and the just received, actual position. Resolving these two states in a believable way can be quite complex. One approach is to create a curve (ex cubic Bézier splines, Catmull-Rom splines, and Hermite curves)[6] between the two states while still projecting into the future. Another technique is to use projective velocity blending, which is the blending of two projections (last known and current) where the current projection uses a blending between the last known and current velocity over a set time.[4]

$V_b = V_0 + \left (\acute{V}_0 - V_0 \right)\hat{T}$
$P_t = P_0 + V_bT_t + \frac{1}{2}\acute{A}_0T_t^2$
$\acute{P}_t = \acute{P}_0 + \acute{V}_0T_t + \frac{1}{2}\acute{A}_0T_t^2$
$Pos = P_t + \left (\acute{P}_t - P_t \right)\hat{T}$

## References

1. ^ Gallistel. The Organization of Learning. 1990.
2. ^ Dead reckoning (path integration) requires the hippocampal formation: evidence from spontaneous exploration and spatial learning tasks in light (allothetic) and dark (idiothetic) tests, IQ Whishaw, DJ Hines, DG Wallace, Behavioural Brain Research 127 (2001) 49 – 69
3. ^ http://www.irbs.com/bowditch/pdf/chapt07.pdf
4. ^ a b Murphy, Curtiss. Believable Dead Reckoning for Networked Games. Published in Game Engine Gems 2, Lengyel, Eric. AK Peters, 2011, p 308-326.
5. ^ Van Verth, James. Essential Mathematics for Games And Interactive Applications. Second Edition. Morgan Kaufmann, 1971, p. 580.
6. ^ Lengyel, Eric. Mathematics for 3D Game Programming And Computer Graphics. Second Edition. Charles River Media, 2004.