Inertial navigation system

Inertial navigation is a self-contained navigation technique in which measurements provided by accelerometers and gyroscopes are used to track the position and orientation of an object relative to a known starting point, orientation and velocity. Inertial measurement units (IMUs) typically contain three orthogonal rate-gyroscopes and three orthogonal accelerometers, measuring angular velocity and linear acceleration respectively. By processing signals from these devices it is possible to track the position and orientation of a device.

Inertial navigation is used in a wide range of applications including the navigation of aircraft, tactical and strategic missiles, spacecraft, submarines and ships. It is also embedded in the mostly nowadays mobile phone for purpose of mobile phone location and tracking [5][6] Recent advances in the construction of microelectromechanical systems (MEMS) have made it possible to manufacture small and light inertial navigation systems. These advances have widened the range of possible applications to include areas such as human and animal motion capture.

An inertial navigation system includes at least a computer and a platform or module containing accelerometers, gyroscopes, 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.) accompanied with the initial orientation 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 acceleration and angular velocity applied to the system. Since it requires no external reference (after initialization), it is immune to jamming and deception.

Inertial navigation systems are used in many different moving objects. However, their cost and complexity place constraints on the environments in which they are practical for use.

Gyroscopes measure the angular velocity of the sensor frame with respect to the inertial reference frame. By using the original orientation of the system in the inertial reference frame as the initial condition and integrating 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, the passenger 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 moving vehicle in the sensor or body 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 their 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 accelerating relative to itself, that is, whether it is accelerating 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 they felt the accelerations.

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 kinematic equations 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, then 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 are compounded into still greater errors in position.[7][8] Since the new position is calculated from the previous calculated position and the measured acceleration and angular velocity, these errors accumulate roughly proportionally to the time since the initial position was input. Even the best accelerometers, with a standard error of 10 micro-g, would accumulate a 50-meter error within 17 minutes.[9] Therefore, the position must be periodically corrected by input from some other type of navigation system.

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 velocity update. In aerospace particularly, other measurement systems are used to determine INS inaccuracies, e.g. the Honeywell LaseRefV inertial navigation systems uses GPS and air data computer outputs to maintain required navigation performance. The navigation error rises with the lower sensitivity of the sensors used. Currently, devices combining different sensors are being developed, e.g. attitude and heading reference system. Because the navigation error is mainly influenced by the numerical integration of angular rates and accelerations, the Pressure Reference System was developed to use one numerical integration of the angular rate measurements.

Estimation theory in general and Kalman filtering in particular,[10] provide a theoretical framework for combining information from various sensors. One of the most common alternative sensors is a satellite navigation radio such as GPS, which can be used for all kinds of vehicles with direct sky visibility. Indoor applications can use pedometers, distance measurement equipment, or other kinds of position sensors. By properly combining the information from an INS and other systems (GPS/INS), the errors in position and velocity are stable. Furthermore, INS can be used as a short-term fallback while GPS signals are unavailable, for example when a vehicle passes through a tunnel.

In 2011, GPS jamming at the civilian level became a governmental concern.[11] The relative ease in ability to jam these systems has motivated the military to reduce navigation dependence on GPS technology.[12] Because inertial navigation sensors do not depend on radio signals unlike GPS, they cannot be jammed.[13] In 2012, researchers at the U.S. Army Research Laboratory reported an inertial measurement unit consisting of micro-electromechanical system triaxial accelerometers and tri-axial gyroscopes with an array size of 10 that had a Kalman filter algorithm to estimate sensor nuisance parameters (errors) and munition position and velocity.[12] Each array measures six data points and the system coordinates the data together to deliver a navigation solution. If one sensor consistently over or underestimates distance, the system can adjust, adjusting the corrupted sensor's contributions to the final calculation.[14]

The addition of the heuristic algorithm reduced a flight's calculated distance error from 120m to 40m from the designated target. The researchers coupled the algorithm with GPS or radar technology to initial and aid the navigation algorithm. At various points during the munition's flight they would cut off tracking and estimate the accuracy of the munition's landing. In a forty-second flight, 10s and 20s availability of aiding demonstrated little difference in error as both were approximately 35m off target. No noticeable difference was observed when experimentation took place with 100 sensor arrays rather than ten.[12] The researchers indicate this limited experimental data signifies an optimization of navigation technology and a potential reduction in cost of military systems.[14]

Inertial navigation systems were originally developed for rockets. American rocketry pioneer Robert Goddard experimented with rudimentary gyroscopic systems. Dr. Goddard's systems were of great interest to contemporary German pioneers including Wernher von Braun. The systems entered more widespread use with the advent of spacecraft, guided missiles, and commercial airliners.

Early German World War II V2 guidance systems combined 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 graphite rudders in the rocket exhaust for flight control. The GN&C (Guidance, Navigation, and Control) system for the 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 under the provisions of Operation Paperclip and were subsequently moved to Huntsville, Alabama, in 1950[15] where they worked for U.S. Army rocket research programs.

In the early 1950s, the US government wanted to insulate itself against over dependency on the German team for military applications, including the development of a fully domestic missile guidance program. 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 [16][17][18][19] (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. 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 and 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 von Braun at Peenemünde 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 (see Q-guidance) 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 cross-product steering. 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.

In February 1961 NASA awarded MIT a contract for preliminary design study of a guidance and navigation system for Apollo. MIT and the Delco Electronics Div. of General Motors Corp. were awarded the joint contract for design and production of the Apollo Guidance and Navigation systems for the Command Module and the Lunar Module. Delco produced the IMUs (Inertial Measurement Units) for these systems, Kollsman Instrument Corp. produced the Optical Systems, and the Apollo Guidance Computer was built by Raytheon under subcontract (see Apollo on-board guidance, navigation and control system, Dave Hoag, International Space Hall of Fame Dedication Conference in Alamogordo, N.M., October 1976[20]).

For the Space Shuttle, open loop (no feedback) guidance was used to guide the Shuttle from lift-off until Solid Rocket Booster (SRB) separation. After SRB separation the primary Space Shuttle guidance is named PEG (Powered Explicit Guidance). PEG takes into account both the Q system and the predictor-corrector attributes of the original "Delta" System (PEG Guidance). Although many updates to the Shuttle's navigation system had taken place over the last 30 years (ex. GPS in the OI-22 build), the guidance core of the Shuttle GN&C system had 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 were formed that touch GN&C as it is a primary interface to "fly" the vehicle.

One example of a popular INS for commercial aircraft was the Delco Carousel, which provided partial automation of navigation in the days before complete flight management systems became commonplace. The Carousel allowed pilots to enter 9 waypoints at a time and then guided the aircraft from one waypoint to the next using an INS to determine aircraft position and velocity. Boeing Corporation subcontracted the Delco Electronics Div. of General Motors to design and build the first production Carousel systems for the early models (-100, -200 and -300) of the 747 aircraft. The 747 utilized three Carousel systems operating in concert for reliability purposes. The Carousel system and derivatives thereof were subsequently adopted for use in many other commercial and military aircraft. The USAF C-141 was the first military aircraft to utilize the Carousel in a dual system configuration, followed by the C-5A which utilized the triple INS configuration, similar to the 747. The KC-135 fleet was fitted with a dual Carousel system that was aided by a Doppler radar. ARINC Characteristic 704 defines the INS used in commercial air transport.

Inertial navigation unit of French IRBM S3.

INSs contain Inertial Measurement Units (IMUs) which have angular and linear accelerometers (for changes in position); some IMUs include a gyroscopic element (for maintaining an absolute angular reference).

Angular accelerometers measure how the vehicle is rotating in space. Generally, there is 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[21] 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 (x,y,z and θ_x, θ_y and θ_z), it integrates over time the sensed acceleration, together with an estimate of gravity, to calculate the current velocity. Then it integrates the velocity to calculate the current position.

Inertial guidance is difficult without computers. The desire to use inertial guidance in the Minuteman missile and Project Apollo drove 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 surface of the earth must incorporate Schuler tuning so that its platform will continue pointing towards the center of the earth as a vehicle moves from place to place.

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 NED frame of reference.

Where: f is specific force, ${\displaystyle \omega }$ is angular rate, a is acceleration, R is position, ${\displaystyle {\dot {R}}}$ and V are velocity, ${\displaystyle \Omega }$ is the angular velocity of the earth, g is the acceleration due to gravity, ${\displaystyle \Phi ,\lambda }$ 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.[31]

• Adam Air Flight 574
• Attitude control
• Dead reckoning
• Fibre-optic gyroscope
• Globus navigation system
• Guidance system
• Hemispherical resonator gyroscope
• Kalman filter
• Korean Air Lines Flight 007
• LN-3 inertial navigation system
• PIGA accelerometer
• Quantum compass
• Rate integrating gyroscope
• Ring laser gyroscope
• Schuler tuning
• Spacecraft

• Zanetti, Renato; d'Souza, Christopher (2020), "Inertial Navigation", Encyclopedia of Systems and Control, pp. 1–7, doi:10.1007/978-1-4471-5102-9_100036-1, ISBN 978-1-4471-5102-9
• A.D. King (1998). "Inertial Navigation – Forty Years of Evolution" (PDF). GEC Review. General Electric Company plc. 13 (3): 140–149.
• E. v. Hinueber (iMAR Navigation) (2011). "Design of an Unaided Aircraft Attitude Reference System with Medium Accurate Gyroscopes for Higher Performance Attitude Requirements". Inertial Sensors and Systems - Symposium Gyro Technology, Karlsruhe / Germany. iMAR Navigation / DGON. 2011.

• Principle of operation of an accelerometer
• Overview of inertial instrument types
• Oxford Technical Solutions Inertial Navigation Guide
• Listing of open-source Inertial Navigation system
• Impact of inertial sensor errors on Inertial Navigation System position and attitude errors
• Introduction to Inertial Navigation Systems in UAV/Drone Applications