Schuler tuning
Encyclopedia
Schuler tuning is a modification to the electronic control system used in inertial navigation system
s that accounts for the curvature of the Earth. An inertial navigation system, used in submarines, ships, aircraft, and other vehicles to keep track of position, determines directions with respect to three axes pointing 'north', 'east', and 'down'. To detect the vehicle's orientation, the system contains an 'inertial platform' mounted on gimbal
s, with gyroscope
s that keep it pointing in a fixed orientation in space. However, the directions 'north', 'east' and 'down' change as the vehicle moves on the curved surface of the Earth. Schuler tuning describes the modifications necessary to an inertial navigation system to keep the inertial platform always pointing 'north', 'east' and 'down', so it gives correct directions on Earth.
whose period exactly equals the orbit
al period of a hypothetical satellite
orbiting just above the surface of the Earth
(about 84 minutes) will tend to remain pointing at the
center of the Earth when its support is suddenly displaced. Such a pendulum would have a length equal to the radius of the Earth. Consider a simple gravity pendulum, whose length equals the radius
of the Earth, suspended in a uniform gravitational field of the same strength as that experienced at the Earth's surface. If suspended from the surface of the Earth, the bob of the pendulum would be at the center of the Earth. If it is hanging motionless and its support is moved sideways, the bob tends to remain motionless, so the pendulum always points at the center of the Earth. If such a pendulum were attached to the inertial platform of an inertial navigation system, the platform would remain level, facing 'North', 'East', and 'Down', as it was moved about on the surface of the Earth.
is modified to make the platform behave as if it were attached to a pendulum. The inertial platform is mounted on gimbal
s, and an electronic
control system
keeps it pointed in a constant direction with respect to the three axes. As the vehicle moves, the gyroscopes detect changes in orientation, and a feedback loop applies signals to torquers to rotate the platform on its gimbals to keep it pointed along the axes.
To implement Schuler tuning, the feedback loop is modified to tilt the platform as the vehicle moves in the north-south and east-west directions, to keep the platform facing 'down'. To do this, the torquers that rotate the platform are fed a signal proportional to the vehicle's north-south and east-west velocity
. The turning rate of the torquers is equal to the velocity divided by the radius of the Earth R:
So:
The acceleration a is a combination of the actual vehicle acceleration and the acceleration due to gravity acting on the tilting inertial platform. So this equation can be seen as a version of the equation for a simple gravity pendulum
with a length equal to the radius of the Earth. The inertial platform acts as if it were attached to such a pendulum.
Schuler's time constant has other applications. Suppose a tunnel is dug from one end of the earth to the other end straight through the center of earth, a stone dropped in such a tunnel oscillates with Schuler's time constant. It can also be proved that the time is the same constant for a tunnel which is not through the center of earth also.
Inertial navigation system
An inertial navigation system is a navigation aid that uses a computer, motion sensors and rotation sensors to continuously calculate via dead reckoning the position, orientation, and velocity of a moving object without the need for external references...
s that accounts for the curvature of the Earth. An inertial navigation system, used in submarines, ships, aircraft, and other vehicles to keep track of position, determines directions with respect to three axes pointing 'north', 'east', and 'down'. To detect the vehicle's orientation, the system contains an 'inertial platform' mounted on gimbal
Gimbal
A gimbal is a pivoted support that allows the rotation of an object about a single axis. A set of two gimbals, one mounted on the other with pivot axes orthogonal, may be used to allow an object mounted on the innermost gimbal to remain immobile regardless of the motion of its support...
s, with gyroscope
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...
s that keep it pointing in a fixed orientation in space. However, the directions 'north', 'east' and 'down' change as the vehicle moves on the curved surface of the Earth. Schuler tuning describes the modifications necessary to an inertial navigation system to keep the inertial platform always pointing 'north', 'east' and 'down', so it gives correct directions on Earth.
Principle
As first explained by German engineer Maximilian Schuler in a 1923 paper, a pendulumPendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
whose period exactly equals the orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
al period of a hypothetical satellite
Satellite
In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
orbiting just above the surface of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
(about 84 minutes) will tend to remain pointing at the
center of the Earth when its support is suddenly displaced. Such a pendulum would have a length equal to the radius of the Earth. Consider a simple gravity pendulum, whose length equals the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of the Earth, suspended in a uniform gravitational field of the same strength as that experienced at the Earth's surface. If suspended from the surface of the Earth, the bob of the pendulum would be at the center of the Earth. If it is hanging motionless and its support is moved sideways, the bob tends to remain motionless, so the pendulum always points at the center of the Earth. If such a pendulum were attached to the inertial platform of an inertial navigation system, the platform would remain level, facing 'North', 'East', and 'Down', as it was moved about on the surface of the Earth.
Application
Obviously, a pendulum the length of the Earth's radius is impractical, so Schuler tuning doesn't use physical pendulums. Instead, the electronic control system of the inertial navigation systemInertial navigation system
An inertial navigation system is a navigation aid that uses a computer, motion sensors and rotation sensors to continuously calculate via dead reckoning the position, orientation, and velocity of a moving object without the need for external references...
is modified to make the platform behave as if it were attached to a pendulum. The inertial platform is mounted on gimbal
Gimbal
A gimbal is a pivoted support that allows the rotation of an object about a single axis. A set of two gimbals, one mounted on the other with pivot axes orthogonal, may be used to allow an object mounted on the innermost gimbal to remain immobile regardless of the motion of its support...
s, and an electronic
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
keeps it pointed in a constant direction with respect to the three axes. As the vehicle moves, the gyroscopes detect changes in orientation, and a feedback loop applies signals to torquers to rotate the platform on its gimbals to keep it pointed along the axes.
To implement Schuler tuning, the feedback loop is modified to tilt the platform as the vehicle moves in the north-south and east-west directions, to keep the platform facing 'down'. To do this, the torquers that rotate the platform are fed a signal proportional to the vehicle's north-south and east-west velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
. The turning rate of the torquers is equal to the velocity divided by the radius of the Earth R:
So:
The acceleration a is a combination of the actual vehicle acceleration and the acceleration due to gravity acting on the tilting inertial platform. So this equation can be seen as a version of the equation for a simple gravity pendulum
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
with a length equal to the radius of the Earth. The inertial platform acts as if it were attached to such a pendulum.
Schuler's time constant has other applications. Suppose a tunnel is dug from one end of the earth to the other end straight through the center of earth, a stone dropped in such a tunnel oscillates with Schuler's time constant. It can also be proved that the time is the same constant for a tunnel which is not through the center of earth also.