Abstract

The appearance and development of inertial navigation facilitated airplanes to make long flights without relying on ground-based radio signals. Inertial navigation systems have become the basis for guiding missiles and other military systems, providing high accuracy in conditions of suppression of radio signals. Inertial systems do not require external data sources, which makes them especially valuable in non-coupling environments. Inertial systems have become essential for submarines and ships, allowing them to determine their position without going to the surface. Thus, inertial navigation has become the basis for many technologies, significantly improving their functionality and reliability.

               This article discusses the basic principles of an inertial navigation system (INS).

The sections will cover Introduction to INS, INS Structure Chart, Advantages and Disadvantages of INS, and Inertial Labs INS. The conclusion will summarize the benefits of using INS for navigation.

Section 1. Introduction to INS

An inertial navigation system (INS) is a navigation device that uses acceleration sensors (accelerometers), angular velocity sensors (gyroscopes), and a computing device to continuously calculate the angles of orientation, the speed of a moving object without the need for external landmarks [1].

At the same time, there are several types of navigation systems; the most popular are:

• Typically, a gimbal platform allows the gyroscopes to remain in a fixed position relative to the ground, Figure 1 a. This is the traditional approach to inertial navigation. Angle and acceleration are calculated using mechanical or optical gyroscopes, which have their axes fixed in space. It is usually more complex and expensive due to the mechanical structure of the gimbal suspension. They are used in aviation and maritime navigation, where high accuracy is required.
• SINS (Strapdown INS). Gyroscopes and accelerometers are fixed directly to the body of a moving object (such as an airplane or car) without a gimbal, Figure 1 b. It uses algorithms that convert data from inertial sensors into a coordinate system of the moving object, which allows for more efficient use of information and simplifies the design. They are generally less complex and more compact, making them more accessible for various applications. They are widely used in military, civil aircraft, and modern vehicles due to their compactness and flexibility.

a	b

Figure 1. Types of inertial navigation systems: a – INS; b – SINS.

               It is worth noting that in this article, the principle of operation will be considered for SINS. Still, the abbreviation INS will be used for simplicity when referring to the inertial navigation system.

The history of inertial navigation covers several vital stages:

1. Early developments (19th century). The foundations of inertial navigation were laid in works on mechanics and physics. The first ideas about gyroscopes and accelerometers began to form at this time [2, 3].
2. The first gyroscopes (early 20th century). In the early 1900s, the first gyroscopes were used in navigation, but they were bulky and inaccessible for practical use [2, 3].
3. Military application (1940s). During the Second World War, inertial navigation systems began to be actively developed for use in aviation and the Navy. This time was marked by the appearance of the first integrated systems [2, 3].
4. Evolution of technology (1950s-1960s). In the 1950s and 60s, inertial navigation systems became more compact and accurate with the development of electronics. Electronic gyroscopes and accelerometers increased reliability [2, 3].
5. Space Age (1960s-1970s). INS has become critical for space programs such as the Apollo missions. Using inertial systems for navigation in space provided high accuracy [2, 3].
6. Modern systems (1980s – present). Advances in MEMS (microelectromechanical systems) have made it possible to create miniature inertial navigation systems that are used in civil aviation, automobiles, and mobile devices, as shown in Figure 1 b. Modern systems are often combined with GPS to improve accuracy.

Thus, inertial navigation has come a long way, from mechanical devices to modern high-precision systems, which play a crucial role in various fields. Now, let’s move on to the main components of INS.

1.1. Main Components of INS

The main components of INS are gyroscopes and accelerometers, and MEMS sensors are often used today because they are small, lightweight, and inexpensive [4]. A triaxial gyroscope and a triaxial accelerometer are commonly used to determine the angular velocity ω_x,y,z and the linear accelerations a_x,y,z along all three axes of an object, Figure 2. Gyroscopes and accelerometers are often combined into an Inertial Measurement Unit (IMU) [5].

Figure 2. Location of gyroscopes and accelerometers in the IMU.

To understand how an accelerometer works, let’s look at the fundamental equation of inertial navigation (1).

(1)

where is the apparent acceleration; – acceleration of the object’s movement; – gravitational acceleration.

               Consider Figure 3, which conventionally shows the accelerometer. Acceleration acts along the X-axis. As a result, the inert mass moves along the axis of sensitivity of the accelerometer and the X-axis.

Figure 3. How MEMS accelerometer works.

               In this case, according to equation (1), because of the projection of gravitational acceleration on the axis X = 0. The corresponding structure and MEMS circuit capacitively perceive the physical displacement of the inertial mass caused by acceleration; in simple words, when the mass is moved, the capacitance of the capacitor, which is the mass displacement sensor, changes, resulting in a change in the signal at the accelerometer output.

               Now, let’s consider the principle of operation of a gyroscope. Figure 4 conventionally shows a gyroscope.  The mass m oscillates forcibly along the X-axis. These forced oscillations have no effect along the Y-axis if no angular rotation exists around the Z-axis. The corresponding MEMS structure and circuit capacitively sense the physical displacement caused by the Coriolis effect. It can also be a capacitive sensor – a capacitor.

Figure 4. The principle of operation of the MEMS gyroscope.

Thus, we can know about an object’s linear accelerations and angular velocities thanks to gyroscopes and accelerometers. Now, let’s look at how this data is used to determine the orientation and navigation of the object.

1.2 How INS works

               Above, we have familiarized ourselves with the basic principles of operation of inertial sensors, and now we have the angular velocities and linear accelerations of the object at our disposal.

               Let’s consider the principle of INS operation using the example of Figure 5, which shows a car [6]. Due to uneven roads, it can go forward/backward, turn left/right, bend, and change altitude relative to the starting point. To begin with, let’s imagine that a car can only go forward, i.e., its movement occurs in one dimension.

Figure 5. Car.

               It is known from the laws of physics that acceleration is the second derivative of the direction of motion, i.e., of coordinate (2).

(2)

               To obtain the velocity and acceleration coordinates, (3) is integrated.

(3)

Unlike the one-dimensional case, the two-dimensional case requires additional calculations. To determine the vehicle’s position in the geographic coordinate system, i.e., on Earth, the accelerations measured by the accelerometers must be in the same frame of reference as the Earth’s coordinate system. To do this, it is necessary to convert the accelerations into a geographical coordinate system (4).

Figure 6. Transformation between a local frame (car) and a navigation frame (E, N).

In this case, it is necessary to monitor both the forward movement of the vehicle along the X and Y axes and its rotational motion resulting from a change in direction (heading).

(4)

Two accelerometers are used to detect acceleration in two directions. One gyroscope is required to detect rotational motion in a direction perpendicular to the plane of motion. As can be seen, to obtain accelerations in the navigation coordinate system , , heading angle () information is needed to obtain accelerations in the navigation coordinate system. This can be obtained by integrating the gyroscope signal (5).

(5)

               Information about the initial heading must be required to calculate the current rate. Considering headings (5) and accelerations (4), we obtain velocities (6) and coordinates (7) in the same way as for the one-dimensional case.

	(6)
	(7)

               Figure 7. As noted, we need to set initial conditions to solve equations, such as initial heading , velocities , and , as well as coordinates and .

Now, let’s look at the operation of gyroscopes and accelerometers in a three-dimensional coordinate system. Figure 8 shows the triad of accelerometers for two cases: a is no tilt, and b is tilted around the X-axis by the “Pitch” angle.

a	b

Figure 8. Triad of accelerometers: a – rotation angles = 0; the accelerometer is rotated to the “Pitch” angle.

               For the first case, Figure 8 a, the accelerations will be:

               For the second case, Figure 8 b, the accelerations will be:

               In this case, projections of the gravitational acceleration vector will act g on the axis Y and Z.

               Now consider a gyroscope triad on the ground at an arbitrary point P, Figure 9.

Figure 9. The triad of gyroscopes on the Earth’s surface at an arbitrary point P. The axes of the gyroscopes coincide with the axes of the geographic coordinate system.

Define the x-axis as East, the y-axis as North, and the z-axis as Upward. The gyroscopes are stationary. Then, in this case, the angular velocities will be equal to:

Where Earth’s rotation speed (7.29e-5 rad/sec), is the latitude of the place.

In the case of a stationary object, in our case, a car located at point P, whose axes are aligned with a geographical coordinate system, gyroscopes installed in the vehicle will measure the speed of the Earth’s rotation.

For a moving car with velocities, the angular velocity values will be equal to [6]:

where is the radius of curvature of the earth’s ellipsoid in the plane of the meridian; – radius of curvature of the earth’s ellipsoid in the plane of the first vertical (perpendicular to the plane of the meridian); – the height of the object.

In the above examples, for the sake of simplicity, it was assumed that the INS coordinate system (b) is aligned with the navigation coordinate system (n). However, b in actual conditions can be in any arbitrary position relative to n since accelerometers and gyroscopes are mounted on an object that can have any orientation relative to n. In many cases, INS is turned on when the object is stationary, and then velocities = 0; all that remains is to enter the heading, orientation angles, and coordinates. The INS calculates the heading angle and the orientation angles at the initial setpoint. The operator can manually or automatically set coordinates if GPS uses INS.

After determining the initial orientation angles, the algorithm creates a rotation matrix that converts the data from b to n. The rotational speeds measured by the gyroscopes constantly update this matrix. After the transformation, double integration of the acceleration will displace the object’s position relative to the starting point.

Section 2. INS Structure Diagram

               In the previous section, we discussed the basic principles of gyroscopes and accelerometers for determining the orientation of an object. This section will look at INS structural diagrams to better understand how all system components work together and what other measuring devices are used to improve accuracy [6, 7].

               Figure 10 shows the structure diagram of the INS. The accelerometer measurements are made in the body frame (b). A transformation matrix is calculated using the inertial rotation information measured by the gyroscopes, which is then used to convert the accelerometer data into a navigation frame (n).

Figure 10. INS Structure Diagram.

The Earth’s gravitational acceleration is calculated using the corresponding model of gravity found in the computer [8]. The resulting acceleration of the vehicle in n is then obtained by algebraic summation of these two accelerations. Using the double integration, the velocity and position are calculated. Since the two integrations are involved, the initial values of the velocities and position are determined at the initial setting.

This is the basic computation process in an inertial frame of reference, while several consecutive steps will be required to perform the calculations. The ratio is derived from the transformation matrix, while the velocities of the body are directly available from the gyroscope measurement.

               Experience with such systems shows that over time, there is an increase in the errors in determining angles, velocities, and coordinates – this is the main disadvantage of INS. Errors occur due to drifts inherent in MEMS sensors [9].

The most popular solution to this problem is integrating the INS satellite navigation system, Figure 11. GPS, in this case, stands for a receiver with an antenna.

Figure 11. Open circuit of INS and GPS integration.

               INS complexing with GPS is usually performed using a Kalman filter [7, 10]. The input signal for the Kalman filter is the difference in coordinates and velocity components generated by INS and GPS. This difference corresponds to the difference between INS and GPS errors. Passing through the Kalman filter, the error difference is practically cleared of GPS errors. Thanks to the filter capabilities, the output not only estimates coordinate and velocity errors but also estimates errors in the orientation angles of the object. The INS error estimates are then subtracted from the actual INS outputs, and the navigation parameters generated by the INS are practically accurate.

               The disadvantage of such a scheme is that the corrective signal disappears when the GPS signal disappears due to tunnel interference, etc. With it, the positive effect of integration disappears.

               In the diagram shown in Figure 12, error scores are used to generate corrective actions in INS.

Figure 12. Closed-loop integration of INS and GPS.

Using a closed circuit makes it possible to use simpler first- and second-order filters to estimate errors rather than the Kalman filter, like those used to smooth out random interference.

In addition to gyroscopes and accelerometers, the INS is often supplemented by a barometer, magnetometers, and velocity measuring devices such as an odometer, as well as a camera, LiDAR, Doppler Velocity Log (DVL), and the like. [11].

Magnetometers measure the direction relative to the Earth’s magnetic north by observing the local magnetic field. To convert the compass reading into the proper direction to the north, it is necessary to consider the angle of declination, which is the difference between geographic and magnetic north [12]. This angle varies depending on the location, so it is necessary to know where the compass is to accurately determine the direction of the geographical north. However, a magnetic compass has a drawback: the local magnetic field can be distorted near power lines or metal structures such as bridges and buildings. This can lead to significant and unpredictable heading errors, making this vehicle navigation system unreliable.

Odometry data is obtained using sensors that measure the rotation of the wheel axles and steering. The curved distance is calculated based on this rotation data. The advantages of this method include high short-term accuracy and low cost. The main disadvantages are that any minor permanent errors accumulate, and orientation errors can lead to significant errors in position determination that increase with the distance traveled. These pros and cons of odometry show that its characteristics complement GPS data well, which is why it is often integrated into GPS+INS systems.

A barometer is used to measure atmospheric pressure, which allows you to estimate altitude above sea level. This is especially useful in aviation and ground navigation, where accurate altitude determination is critical. In combination with GPS and inertial sensors, the barometer improves the overall accuracy of the navigation system. However, there are also disadvantages. Atmospheric pressure can vary due to weather changes, leading to errors in altitude measurements. Differential pressure sensors are also used to estimate wind speed.

Section 3. Advantages and disadvantages of INS

Let’s take a look at the advantages and disadvantages of INS.

Advantages of INS:

1. Autonomy. INS do not depend on external sources such as satellites or beacons. This allows the system to be used in environments where such signals are unavailable (e.g., underwater, in tunnels).
2. High reaction speed. INS can quickly update position and speed data, which is especially important for dynamic objects such as airplanes or missiles.
3. Accuracy in the short term. In the initial period of operation, the system can provide high navigation accuracy, primarily if the initial conditions are known.
4. Ability to integrate. INS can be combined with other navigation systems (such as GPS) to improve overall reliability and accuracy.
5. No external interference. The system is not subject to interference from external sources such as atmospheric conditions or electromagnetic interference.

Disadvantages of INS:

1. Accumulation of errors. The main disadvantage of INS is the accumulation of errors. Measurement errors in inertial sensors (gyroscopes and accelerometers) lead to significant deviations from the actual position over time.
2. The need for initial data. INS requires precise initial conditions to provide higher accuracy, which can be challenging to achieve in some situations.
3. Dependence on the quality of sensors. The accuracy of INS depends mainly on the quality of the sensors used. Low-quality sensors can lead to significant errors.

Section 4. Inertial Labs INS

Sensor fusion, the cornerstone of auxiliary inertial navigation systems, combines data from different sensors to achieve a more accurate and reliable assessment of the state of a moving object (position, orientation, and velocity) than would be obtained from any single sensor [13]. The integration process is about averaging data from these sources and involves complex algorithms that understand each data type’s context, accuracy, and reliability. For example, GNSS data can be heavily weighted in open-sky conditions but not accounted for in urban canyons, where signals are prone to reflections and interference. Inertial Labs recently released an improved navigation algorithm that increases navigational accuracy for Ground Vehicles in GNSS-Denied environments. This “Tunnel Guide” feature has been proven through trials and testing to result in a position accuracy of 0.2% over Distance Travelled (DT) in GNSS-denied environments.

In addition, the systems use high-precision gyroscopes and accelerometers with a low drift value, Figure 13. Inertial Labs also provides various magnetometer calibration methods for multiple applications to ensure maximum accuracy regardless of situations with no GNSS signal [14].

Figure 13. IMU-P Parameters.

              Various INS are available, as shown in Figure 14 [15], depending on the users’ needs.

Figure 14. INS models.

A distinctive feature of Inertial Labs systems is their exceptional accuracy. By integrating advanced algorithms and sensor fusion techniques, their products provide highly accurate location, speed, and orientation data in challenging environments. The ability of the systems to incorporate data from a multi-constellation GNSS receiver offers comprehensive global coverage and immunity to signal failures.

Conclusion

               This article considers the principles of operation of the inertial navigation system (INS), its key components, and operation algorithms. INS is based on the measurement of angular and linear accelerations using gyroscopes and accelerometers, which allows you to accurately calculate the position and orientation of an object in space. Due to their autonomy and high accuracy, inertial navigation systems are widely used in aviation, space research, military equipment, and automotive navigation. Advances in technologies such as the miniaturization of sensors and improved data processing algorithms are opening up new horizons for using INS in the future, further improving the accuracy and reliability of navigation in various fields.

Among the many developers, Inertial Labs™ stands out for offering customizable solutions that meet the unique needs of different applications. Their systems are designed to be easily integrated with a wide range of external sensors, allowing for flexibility that improves system performance and simplifies user integration. This approach not only simplifies development but also significantly reduces the associated costs.

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