An Inertial Navigation System
(INS) is a navigation
aid that uses a computer
, motion sensors (accelerometers
) and rotation sensors (gyroscopes
) to continuously calculate via
orientation, and velocity
speed of movement) of a moving object without the need for external
references. It is used on vehicles such as ships
, and spacecraft
. Other terms
used to refer to inertial navigation systems or closely related
devices include inertial guidance system
inertial reference platform
, and many other
An inertial navigation system includes at least a computer and a
platform or module containing accelerometers
, or other motion-sensing devices. The
INS is initially provided with its position and velocity from
another source (a human operator, a GPS satellite receiver, etc.),
and thereafter computes its own updated position and velocity by
integrating information received from the motion sensors. The
advantage of an INS is that it requires no external references in
order to determine its position, orientation, or velocity once it
has been initialized.
An INS can detect a change in its geographic position (a move east
or north, for example), a change in its velocity (speed and
direction of movement), and a change in its orientation (rotation
about an axis). It does this by measuring the linear and angular
applied to the system.
Since it requires no external reference (after initialization), it
is immune to jamming
Inertial-navigation systems are used in many different moving
objects, including vehicles—such as aircraft
—and guided missiles
. However, their cost and
complexity place constraints on the environments in which they are
practical for use.
Gyroscopes measure the angular
of the system in the inertial reference frame
. By using
the original orientation of the system in the inertial reference
frame as the initial condition
the angular velocity, the
system's current orientation is known at all times. This can be
thought of as the ability of a blindfolded passenger in a car to
feel the car turn left and right or tilt up and down as the car
ascends or descends hills. Based on this information alone, he
knows what direction the car is facing but not how fast or slow it
is moving, or whether it is sliding sideways.
Accelerometers measure the linear acceleration of the system in the
inertial reference frame, but in directions that can only be
measured relative to the moving system (since the accelerometers
are fixed to the system and rotate with the system, but are not
aware of their own orientation). This can be thought of as the
ability of a blindfolded passenger in a car to feel himself pressed
back into his seat as the vehicle accelerates forward or pulled
forward as it slows down; and feel himself pressed down into his
seat as the vehicle accelerates up a hill or rise up out of his
seat as the car passes over the crest of a hill and begins to
descend. Based on this information alone, he knows how the vehicle
is moving relative to itself, that is, whether it is going forward,
backward, left, right, up (toward the car's ceiling), or down
(toward the car's floor) measured relative to the car, but not the
direction relative to the Earth, since he did not know what
direction the car was facing relative to the Earth when he felt the
However, by tracking both the current angular velocity of the
system and the current linear acceleration of the system measured
relative to the moving system, it is possible to determine the
linear acceleration of the system in the inertial reference frame.
Performing integration on the inertial accelerations (using the
original velocity as the initial conditions) using the correct
yields the inertial velocities of the system, and integration again
(using the original position as the initial condition) yields the
inertial position. In our example, if the blindfolded passenger
knew how the car was pointed and what its velocity was before he
was blindfolded, and if he is able to keep track of both how the
car has turned and how it has accelerated and decelerated since, he
can accurately know the current orientation, position, and velocity
of the car at any time.
All inertial navigation systems suffer from "integration drift":
small errors in the measurement of acceleration and angular
velocity are integrated into progressively larger errors in
velocity, which is compounded into still greater errors in
position. Since the new position is calculated from the previous
calculated position and the measured acceleration and angular
velocity, these errors are cumulative and increase at a rate
roughly proportional to the time since the initial position was
input. Therefore the position must be periodically corrected by
input from some other type of navigation system. The inaccuracy of
a good-quality navigational system is normally less than 0.6
per hour in position
and on the order of tenths of a degree per hour in
Accordingly, inertial navigation is usually used to supplement
other navigation systems, providing a higher degree of accuracy
than is possible with the use of any single system. For example,
if, in terrestrial use, the inertially tracked velocity is
intermittently updated to zero by stopping, the position will
remain precise for a much longer time, a so-called zero
in general and
provide a theoretical framework for combining information from
various sensors. One of the most common alternative sensors is a
such as GPS
. By properly combining the
information from an INS and the GPS system (GPS/INS
), the errors in position and velocity are
. Furthermore, INS can be
used as a short-term fallback while GPS signals are unavailable,
for example when a vehicle passes through a tunnel.
Inertial navigation systems were originally developed for rockets
. American rocket pioneer Robert Goddard
Goddard's systems were of great interest to contemporary German
pioneers including Wernher von
. The systems entered more widespread use with the advent
, guided missiles
, and commercial airliners
Early German WWII V2 guidance systems
two gyroscopes and a lateral accelerometer with a simple analog
computer to adjust the azimuth for the rocket in flight. Analog
computer signals were used to drive four external rudders on the
tail fins for flight control. The GN&C system for V2 provided
many innovations as an integrated platform with closed loop
guidance. At the end of the war Von Braun engineered the surrender
of 500 of his top rocket scientists, along with plans and test
vehicles, to the Americans. They arrived at Fort Bliss, Texas in
1945 and were subsequently moved to Huntsville, Alabama, in 1950
where they worked for U.S. military rocket research programs.
In the early 1950s, the US government wanted to insulate itself
against over dependency on the Germany team for military
applications. Among the areas that were domestically "developed"
was missile guidance. In the early 1950's the MIT Instrumentation
Laboratory (later to become the Charles Stark Draper Laboratory,
Inc.) was chosen by the Air Force Western Development Division to
provide a self-contained guidance system backup to Convair in San
Diego for the new Atlas intercontinental ballistic missile  
(Construction and testing were
completed by Arma Division of AmBosch Arma). The technical monitor
for the MIT task was a young engineer named Jim Fletcher who later
served as the NASA Administrator. The Atlas guidance system was to
be a combination of an on-board autonomous system, and a
ground-based tracking and command system. This was the beginning of
a philosophic controversy, which, in some areas, remains
unresolved. The self-contained system finally prevailed in
ballistic missile applications for obvious reasons. In space
exploration, a mixture of the two remains.
In the summer of 1952, Dr. Richard Battin and Dr. J. Halcombe
"Hal" Laning, Jr.
, researched computational based solutions to
guidance. Dr. Laning, with the help of Phil Hankins and Charlie
Werner, initiated work on MAC, an algebraic programming language
for the IBM 650
, which was completed by
early spring of 1958. MAC became the work-horse of the MIT lab. MAC
is an extremely readable language having a three-line format,
vector-matrix notations and mnemonic and indexed subscripts.
Today's Space Shuttle (STS) language called HAL/S, (developed by
Intermetrics, Inc.) is a direct offshoot of MAC. Since the
principal architect of HAL was Jim Miller, who co-authored a report
on the MAC system with Hal Laning, it is probable the Space Shuttle
language is named for Laning and not, as some have suggested, for
the electronic rstar
of Stanley Kubrik's
2001: A Space
Hal Laning and Richard Battin undertook the initial analytical work
on the Atlas inertial guidance in 1954. Other key figures at
Convair were Charlie Bossart, the Chief Engineer, and Walter
Schweidetzky, head of the guidance group. Schweidetzky had worked
with Wernher von Braun at Peenemuende during World War II.
The initial Delta guidance system assessed the difference in
position from a reference trajectory. A velocity to be gained (VGO)
calculation is made to correct the current trajectory with the
objective of driving VGO to zero. The mathematics of this approach
were fundamentally valid, but dropped because of the challenges in
accurate inertial guidance and analog computing power. The
challenges faced by the Delta efforts were overcome by the Q system
) of guidance. The Q
system's revolution was to bind the challenges of missile guidance
(and associated equations of motion) in the matrix Q. The Q matrix
represents the partial derivatives of the velocity with respect to
the position vector. A key feature of this approach allowed for the
components of the vector cross product (v, xdv, /dt) to be used as
the basic autopilot rate signals—a technique that became known as
. The Q-system was presented at the
first Technical Symposium on Ballistic Missiles held at the
Ramo-Wooldridge Corporation in Los Angeles on June 21 and 22, 1956.
The Q system was classified information through the 1960s.
Derivations of this guidance are used for today's missiles.
Guidance in Human spaceflight
In Feb of 1961 NASA Awarded MIT a contract for preliminary design
study of a guidance and navigation system for Apollo. Apollo Guidance Computer
by Raytheon under subcontract to MIT/CSDL. (see Apollo on-board
guidance, navigation, and control system ,Dave Hoag, International
Space Hall of Fame Dedication Conference in Alamogordo, N.M.,
October 1976 
For the space shuttle, an open loop (no feedback) guidance is used
to guide the shuttle from lift off until Solid Rocket Booster
(SRB). After SRB separation the primary space shuttle guidance is
named PEG4 (Powered Explicit Guidance). PEG4 takes into account
both the Q system and the predictor-corrector attributes of the
original "Delta" System (PEG Guidance). Although many updates to
the shuttles navigation system have taken place over the last 30
years (ex. GPS in the OI-22 build), the guidance core of today's
Shuttle GN&C system has evolved little. Within a manned system,
there is a human interface needed for the guidance system. As
Astronauts are the customer for the system, many new teams are
formed that touch GN&C as it is a primary interface to "fly"
Aircraft inertial guidance
One example of a popular INS for commercial aircraft was the
, which provided
partial automation of navigation in the days before complete
flight management systems
became commonplace. The Carousel allowed pilots to enter a series
of waypoints, and then guided the aircraft from one waypoint to the
next using an INS to determine aircraft position. Some aircraft
were equipped with dual Carousels for safety.
Inertial navigation systems in detail
INSs have angular and linear accelerometers (for changes in
position); some include a gyroscopic element (for maintaining an
absolute angular reference).
Angular accelerometers measure how the vehicle is rotating in
space. Generally, there's at least one sensor for each of the three
axes: pitch (nose up and down), yaw (nose left and right) and roll
(clockwise or counter-clockwise from the cockpit).
Linear accelerometers measure non-gravitational accelerations of
the vehicle. Since it can move in three axes (up & down, left
& right, forward & back), there is a linear accelerometer
for each axis.
A computer continually calculates the vehicle's current position.
First, for each of the six degrees of freedom
), it integrates
over time the sensed amount of acceleration, together with an
estimate of gravity, to calculate the current velocity. Then it
integrates the velocity to figure the current position.
Inertial guidance is difficult without computers. The desire to use
inertial guidance in the Minuteman
and Project Apollo
early attempts to miniaturize computers.
Inertial guidance systems are now usually combined with satellite navigation systems
through a digital filtering system. The inertial system provides
short term data, while the satellite system corrects accumulated
errors of the inertial system.
An inertial guidance system that will operate near the surfaceof
the earth must incorporate Schuler
so that itsplatform will continue pointing towards the
center of the earthas a vehicle moves from place to place.
Gimballed gyrostabilized platforms
Some systems place the linear accelerometers on a gimbaled
gyrostabilized platform. The gimbals
set of three rings, each with a pair of bearings initially at right
angles. They let the platform twist about any rotational axis (or,
rather, they let the platform keep the same orientation while the
vehicle rotates around it). There are two gyroscopes
(usually) on the platform.
Two gyroscopes are used to cancel gyroscopic
, the tendency of a gyroscope to twist at right
angles to an input force. By mounting a pair of gyroscopes (of the
same rotational inertia and spinning at the same speed) at right
angles the precessions are cancelled, and the platform will resist
This system allows a vehicle's roll, pitch, and yaw angles to be
measured directly at the bearings of the gimbals. Relatively simple
electronic circuits can be used to add up the linear accelerations,
because the directions of the linear accelerometers do not
The big disadvantage of this scheme is that it uses many expensive
precision mechanical parts. It also has moving parts
that can wear out or jam, and is
vulnerable to gimbal lock
. The primary guidance system
of the Apollo spacecraft
used a three-axis
gyrostabilized platform, feeding data to the Apollo Guidance Computer
had to be carefully planned to avoid gimbal lock.
Fluid-suspended gyrostabilized platforms
Gimbal lock constrains maneuvering, and it would be beneficial to
eliminate the slip rings and bearings of the gimbals. Therefore,
some systems use fluid bearings or a flotation chamber to mount a
gyrostabilized platform. These systems can have very high
precisions (e.g., Advanced Inertial Reference
). Like all gyrostabilized platforms, this system runs
well with relatively slow, low-power computers.
The fluid bearings are pads with holes through which pressurized
inert gas (such as Helium) or oil press against the spherical shell
of the platform. The fluid bearings are very slippery, and the
spherical platform can turn freely. There are usually four bearing
pads, mounted in a tetrahedral arrangement to support the
In premium systems, the angular sensors are usually specialized
coils made in a strip on a flexible printed circuit board
. Several coil
strips are mounted on great circles
around the spherical shell of the gyrostabilized platform.
Electronics outside the platform uses similar strip-shaped
transformers to read the varying magnetic fields produced by the
transformers wrapped around the spherical platform. Whenever a
magnetic field changes shape, or moves, it will cut the wires of
the coils on the external transformer strips. The cutting generates
an electric current in the external strip-shaped coils, and
electronics can measure that current to derive angles.
Cheap systems sometimes use bar codes
sense orientations, and use solar cells
or a single transformer to power the platform. Some small missiles
have powered the platform with light from a window or optic fibers
to the motor. A research topic is to suspend the platform with
pressure from exhaust gases. Data is returned to the outside world
via the transformers, or sometimes LEDs
communicating with external photodiodes
Lightweight digital computers permit the system to eliminate the
gimbals, creating strapdown
systems, so called because their sensors are simply strapped to the
vehicle. This reduces the cost, eliminates gimbal lock
, removes the need for some
calibrations, and increases the reliability by eliminating some of
the moving parts. Angular rate sensors called rate gyros
measure how the angular velocity of the vehicle changes.
A strapdown system has a dynamic measurement range several hundred
times that required by a gimbaled system. That is, it must
integrate the vehicle's attitude changes in pitch, roll and yaw, as
well as gross movements. Gimballed systems could usually do well
with update rates of 50–60 Hz. However, strapdown systems
normally update about 2000 Hz. The higher rate is needed to
keep the maximum angular measurement within a practical range for
real rate gyros: about 4 milliradians. Most rate gyros are now
The data updating algorithms (direction cosines
) involved are too complex to
be accurately performed except by digital electronics. However,
are now so
inexpensive and fast that rate gyro systems can now be practically
used and mass-produced. The Apollo lunar
used a strapdown system in its backup Abort Guidance
Strapdown systems are nowadays commonly used in commercial and
tactical applications (aircraft, missiles
etc.). However they are still not widespread in applications where
superb accuracy is required (like submarine
navigation or strategic ICBM
The orientation of a gyroscope system can sometimes also be
inferred simply from its position history (e.g., GPS). This is, in
particular, the case with planes and cars, where the velocity
vector usually implies the orientation of the vehicle body.
For example, Honeywell
's Align in
is an initialization process where the initialization
occurs while the aircraft is moving, in the air or on the ground.
This is accomplished using GPS
and an inertial
reasonableness test, thereby allowing commercial data integrity
requirements to be met. This process has been FAA certified to
recover pure INS performance equivalent to stationary align
procedures for civilian flight times up to 18 hours.It avoids the
need for gyroscope batteries on aircraft.
Less-expensive navigation systems, intended for use in automobiles,
may use a vibrating
to detect changes in heading, and the
odometer pickup to measure distance covered along the vehicle's
track. This type of system is much less accurate than a higher-end
INS, but it is adequate for the typical automobile application
where GPS is the primary navigation system, and dead reckoning
is only needed to fill gaps in
GPS coverage when buildings or terrain block the satellite
Hemispherical resonator gyros (brandy snifter gyros)
If a standing wave is induced in a globular resonant cavity
(e.g., a brandy snifter
), and then the snifter is
tilted, the waves tend to continue oscillating in the same plane of
movement—they don't fully tilt with the snifter. This trick is used
to measure angles. Instead of brandy snifters, the system uses
hollow globes machined from piezoelectric
materials such as quartz
. The electrodes to start and sense the waves
are evaporated directly onto the quartz.
This system has almost no moving parts, and is very accurate.
However it is still relatively expensive due to the cost of the
precision ground and polished hollow quartz spheres. Although
successful systems were constructed, and an HRG's kinematics appear
capable of greater accuracy, they never really caught on. Laser
gyros were just more popular. The classic system is the Delco 130Y
Hemispherical Resonator Gyro, developed about 1986. See also
for a picture of an HRG resonator.
Quartz rate sensors
This system is usually integrated on a silicon chip. It has two
mass-balanced quartz tuning forks, arranged "handle-to-handle" so
forces cancel. Aluminum electrodes evaporated onto the forks and
the underlying chip both drive and sense the motion. The system is
both manufacturable and inexpensive. Since quartz is dimensionally
stable, the system can be accurate.
As the forks are twisted about the axis of the handle, the
vibration of the tines tends to continue in the same plane of
motion. This motion has to be resisted by electrostatic forces from
the electrodes under the tines. By measuring the difference in
capacitance between the two tines of a fork, the system can
determine the rate of angular motion.
Current state of the art non-military technology ( ) can build
small solid state sensors that can measure human body movements.
These devices have no moving parts, and weigh about 50 grams.
Solid state devices using the same physical principles are used to
stabilize images taken with small cameras or camcorders. These can
be extremely small (≈5 mm) and are built with Microelectromechanical
Sensors based on magnetohydrodynamic principles
used to measure angular velocities.
to eliminate the bearings in the gyroscopes, and thus the last
bastion of precision machining and moving parts.
A ring laser gyro splits a beam of laser
into two beams in opposite directions through narrow tunnels in a
closed optical circular path around the perimeter of a triangular
block of temperature-stable cervit glass with reflecting mirrors
placed in each corner. When the gyro is rotating at some angular
rate, the distance traveled by each beam becomes different—the
shorter path being opposite to the rotation. The phase-shift
between the two beams can be measured by an interferometer, and is
proportional to the rate of rotation (Sagnac effect
In practice, at low rotation rates the output frequency can drop to
zero after the result of back
causing the beams to synchronise and lock together.
This is known as a lock-in
, or laser-lock
result is that there is no change in the interference pattern, and
therefore no measurement change.
To unlock the counter-rotating light beams, laser gyros either have
independent light paths for the two directions (usually in fiber
optic gyros), or the laser gyro is mounted on a piezo-electric
dither motor that rapidly vibrates the laser ring back and forth
about its input axis through the lock-in region to decouple the
The shaker is the most accurate, because both light beams use
exactly the same path. Thus laser gyros retain moving parts, but
they do not move as far.
Principle of open loop
Acceleration in the upward direction causes the mass to
The basic, open-loop accelerometer consists of a mass attached to a
spring. The mass is constrained to move only in-line with the
spring. Acceleration causes deflection of the mass and the offset
distance is measured. The acceleration is derived from the values
of deflection distance, mass, and the spring constant. The system
must also be damped to avoid oscillation.A closed-loop
accelerometer achieves higher performance by using a feedback loop
to cancel the deflection, thus keeping the mass nearly stationary.
Whenever the mass deflects, the feedback loop causes an electric
coil to apply an equally negative force on the mass, cancelling the
motion. Acceleration is derived from the amount of negative force
applied. Because the mass barely moves, the non-linearities of the
spring and damping system are greatly reduced. In addition, this
accelerometer provides for increased bandwidth past the natural
frequency of the sensing element.
Both types of accelerometers have been manufactured as integrated
micromachinery on silicon chips.
In one form, the navigational system of equations acquires linear
and angular measurements from the inertial and body frame,
respectively and calculates the final attitude and position in the
frame of reference.
Where:f is specific force, \omega is angular rate, a is
acceleration, R is position, \dot R and V are velocity, \Omega is
the angular velocity of the earth, g is the acceleration due to
gravity, \Phi, \lambda and h are the NED location parameters. Also,
super/subscripts of E, I and B are representing variables in the
Earth centered, Inertial or Body reference frame, respectively and
C is a transformation of reference frames.
- Battin, Richard, AIAA 82-4075, April 1982.
- Eshbach's Handbook of Engineering Fundamentals By Ovid W.
Eshbach, Byron pg 9
- Doug Weed, et al.: GPS Align in Motion of Civilian Strapdown INS.
Honeywell Commercial Aviation Products.
- Sherryl H. Stovall Basic Inertial Navigation, Naval Air Warfare
Center Weapons Division, September 1997