Inertial Navigation Systems (INS) have to take the Gravity Vector, Earth Rotation Rate and the Coriolis Effect into account to be able to correctly calculate the movement and orientation of the aircraft over the globe without any reference to the outside.

What is inertial navigation?

"Inertial navigation is accomplished by integrating the output of a set of sensors to compute position, velocity, and attitude. The sensors used are gyros and accelerometers. Gyros measure angular rate with respect to inertial space, and accelerometers measure linear acceleration, again with respect to an inertial frame. Integration is a simple process, complexities arise due to the various coordinate frames encountered, sensor errors, and noise in the system."

•  Basic Inertial Navigation by Sherryl Stovall.pdf 

"Inertial Navigation (IN) is the type of navigation that uses accelerometers and a computer to calculate via dead reckoning one’s position. IN is fundamentally achieved by the double integration of acceleration and correcting for gravitational fields in the computations along with other considerations such as coriolis acceleration. Currently IN is used for navigation in astronautics, aeronautics, marine and land, but the technology goes beyond navigation"

• Optimization of a Strapdown Inertial Navigation System; Mario Ruiz; University of Texas at El Paso, mariobikes@hotmail.com

See page 23, Navigation Equations

"The navigation computer uses these equations to compute the vehicle’s position taking measurements form the sensors, correcting them to account gravity, coriolis, and the earth’s rotation, and then computing the vehicles movements."

"A moving vehicle experiences forces like Coriolis and centripetal which related to the vehicle’s craft rate. These forces need to be accounted for to accurately navigate." page 65

Introductions to INS

Search for Coriolis, Earth Rotation and Gravity:

•  Basic Inertial Navigation  by Sherryl H. Stovall
• INERTIAL NAVIGATION SYSTEMS; IVAO INTERNATIONAL VIRTUAL AVIATION ORGANISATION
• Inertial Navigation System (INS); SKYbrary

Basic Inertial Navigation by Sherryl H. Stovall

Screenshot from a report, that is intended as a tutorial on inertial navigation. The basic principles are presented with a minimum of mathematical detail. This work was funded by the AIM-9X Program Office.

"One consequence of the earth’s rotation other than increased mathematical complexity exists: If we perform a stationary alignment, we can estimate heading as well as tilts because the earth’s rotation rate sensed by the gyros. This technique is known as gyrocompassing, which can produce very accurate alignments at latitudes below 70 degrees."

Source:  Basic Inertial Navigation by Sherryl Stovall.pdf 

Integration of a GPS aided Strapdown Inertial Navi

The accelerometers Acc are affected by the Coriolis force which depends on the velocity $\dot{\mathbf{r}}$ and earth's rotation ω. The accelerometers also measure the Gravity vector, which is calculated from the known position of the aircraft $\mathbf{r}$ and has to be canceled to get the pure aircraft accelerations.

The gyroscopes Gyros not only measure the rotation of the aircraft but also the rotation of the earth. To get the pure aircraft rotation, earth's rotation has to be subtracted from the rotation rate measured by the gyros.

Source: Integration of a GPS aided Strapdown Inertial Navi

INS Block Diagram Titterton Weston

Inertial navigation system algorithm block diagram. Taken from Titterton & Weston (2004) The navigation equations presented in the image above is collectively known as the INS (Inertial Navigation System). The output of INS consists of the complete system position, velocity, acceleration, orientation and angular rates of rotation. These parameters are used by the control system as the measurements of the system motion on which the control action will be performed.

Source: Design Considerations for Long Endurance Unmanned Aerial Vehicles

Block diagram of the INS working in the navigation coordinate system

The IMU is an underlying element of each INS. Sensors whose output is influenced only by the motion of the object, on which the IMU is placed, are regarded as primary sensors of the IMU. Sensors of angular velocity whose output signals after integrating are used for determining the orientation in space, and accelerometers whose output signals after precise compensating the gravitational acceleration and Coriolis force can be integrated for speed and position are the primary sensors in the inertial navigation. 4 Such an IMU has six degrees of freedom, which means that it enables measuring the translational and rotary motion in three orthogonal axes.

Source: Enhancing the reliability of mobile robots control

Inertial Navigation System (INS) Mechanization

When the accelerometer and gyroscope in the motion sensor measure specific forces and angular rates, respectively, the navigation solutions including position, velocity, and attitude are computed using the inertial navigation system (INS) mechanization.

Source: Performance Evaluation of Autonomous Driving Contr