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Introduction to Inertial Navigation System

Inertial navigation system (referred to as INS) is a system that completes navigation tasks by measuring the acceleration of the carrier itself. It should consist of at least one inertial measurement device, a digital computer, a control display device, and a dedicated precision power supply. According to Newton's principle of inertia, the acceleration of the carrier in the inertial reference frame is measured using inertial elements such as gyroscopes and accelerometers. After integration and calculation, the velocity, attitude angle, and position information in the navigation coordinate system can be obtained for navigation use. Therefore, inertial navigation system is an autonomous navigation system that does not rely on any external information and does not radiate energy to the outside. It has the advantages of high data update rate, high short-term accuracy, and good stability, and is widely used in aerospace, aviation, navigation, and many civilian fields. It has become a major navigation device applied on various types of vehicles.

According to the installation method of inertial measurement devices on the carrier, inertial navigation systems can be divided into:

（1） Platform based inertial navigation system

（2） Strapdown Inertial Navigation System

Due to the fact that the motion of the carrier is carried out in three-dimensional space, there are two forms of motion: linear motion and angular motion. Both linear and angular motion are in three-dimensional space, and to establish a three-dimensional coordinate system, it is necessary to establish a three-axis inertial platform. With a three-axis inertial platform, a benchmark for measuring three degrees of freedom linear acceleration can be provided. Three linear acceleration components with known orientations can be measured, and the motion velocity and position of the carrier can be calculated through a computer. Therefore, the first major type of inertial navigation system solution is a platform based inertial navigation system. Due to the use of complex 'mechanical platforms', their manufacturing and maintenance costs are high, their volume and mass are large, and their reliability is not high. If a "mechanical platform" is not used, the inertial elements gyroscope and accelerometer are directly installed on the carrier, and a "mathematical platform" is established in the computer to obtain the velocity and position of the carrier through complex calculations and transformations. This type of inertial navigation system without a mechanical platform is the second largest type of inertial navigation system solution, called Strapdown Inertial Navigation System (SINS). Their main difference is that platform based inertial navigation systems have an actual physical platform, with gyroscopes and accelerometers placed on a stable platform that tracks the navigation coordinate system to achieve velocity and position calculations, while attitude data is directly taken from the platform's frame; In strapdown inertial navigation, the gyroscope and accelerometer are directly connected to the carrier, and the function of the inertial platform is completed by the computer, which is the so-called "mathematical platform". Due to the lack of a complex platform system framework and servo system for tracking gyroscopes, the strapdown system greatly simplifies the system structure and brings many advantages to the system: the system's volume and cost are greatly reduced; Inertial instruments are easy to install, maintain, and replace; Can provide more navigation information; Inertial instruments facilitate the use of redundant configurations, thereby improving system performance and reliability.

Given the above advantages, strapdown systems have become the main direction for the development of inertial technology. According to reports, the US military's inertial navigation systems were all platform based in 1984, and by 1989, half of them had been converted to strapdown systems. By 1994, strapdown systems had accounted for 90%.

In the 1980s and 1990s, power tuned gyroscopes, laser gyroscopes, and fiber optic gyroscopes were adopted in various civilian fields such as space shuttles, spacecraft, satellites, as well as military fields such as strategic and tactical missiles, military aircraft, anti submarine weapons, and combat ships. Laser gyroscope and fiber optic gyroscope are ideal components for strapdown inertial navigation systems. The strapdown attitude system using fiber optic gyroscope has been used in the onboard weapon system of fighter jets and Boeing 777 aircraft. The average time between failures of the Boeing 777 can reach up to 20000 hours due to the use of a fiber optic gyroscope based inertial navigation system. The strapdown inertial navigation system using fiber optic gyroscope is considered a highly promising navigation system. With the development of aerospace technology and breakthroughs in key technologies of new inertial devices, the reliability and accuracy of strapdown inertial navigation systems will be higher.

Based on the strapdown inertial navigation system, regardless of the accuracy of the inertial components, the errors of gyroscope drift and accelerometer gradually accumulate over time. Long term operation of the inertial navigation system will inevitably lead to objective accumulated errors. Therefore, in addition to continuously exploring ways to improve the accuracy of autonomous inertial navigation systems, people are also seeking to introduce external information to form a combined navigation system, which is an important measure to compensate for the shortcomings of inertial navigation systems. This article will not discuss the combination navigation system.

In addition, inertial navigation systems belong to dead reckoning navigation systems, so there must be an initialization process before navigation. For strapdown inertial navigation systems, initial alignment is to determine the strapdown matrix at the initial time. The initial alignment must be carried out before each start of formal work, and it requires high alignment accuracy and short alignment time. Initial alignment is one of the key technologies for strapdown inertial navigation systems, and it will not be discussed in this article.

Principles of Strapdown Inertial Navigation System

The original meaning of the English term 'Strapdown' is' bundling '. The so-called strapdown inertial navigation system is a system that directly "binds" inertial sensitive elements (gyroscopes and accelerometers) to the body of the carrier to complete navigation tasks.

Common coordinate systems

The coordinate systems used in inertial navigation can be divided into two categories: inertial coordinate systems and non inertial coordinate systems. The fundamental difference between inertial navigation and other types of navigation schemes (such as radio navigation, astronomical navigation, etc.) is that its navigation principle is based on Newton's laws of mechanics (also known as the laws of inertia). However, Newton's laws of mechanics hold in inertial space, so it is necessary to first introduce an inertial coordinate system as a coordinate reference for discussing the basic principles of inertial navigation. The main purpose of navigation is to determine the real-time navigation parameters of the carrier, such as attitude, position, speed, etc. The navigation parameters of the carrier are determined by the relationships between various coordinate systems, which are different from inertial coordinate systems and selected according to the needs of navigation. These coordinate systems are called non inertial coordinate systems, such as the Earth coordinate system, geographic coordinate system, navigation coordinate system, platform coordinate system, and carrier coordinate system.

In inertial navigation, the commonly used coordinate systems are as follows:

1. Geocentric Inertial Coordinate System (i-System) - OeXiYiZi

The geocentric coordinate system is an inertial coordinate system (Figure 1), which is a coordinate system that is absolutely stationary or only in uniform linear motion. The origin Oe of the geocentric coordinate system is selected at the center of the Earth; The Zi axis is selected in the direction pointing towards the North Pole along the Earth's axis, while the Xi and Yi axes are in the Earth's equatorial plane and point towards two stars in space. XiYiZi forms a right-handed coordinate system. The three coordinate axes point to the inertial space and remain fixed, serving as the reference frame for inertial instrument measurement.

Figure 1 Geocentric inertial coordinate system

4. Carrier coordinate system (b system) - OXbYbZb

The carrier coordinates are the coordinate system fixedly attached to the carrier (as shown in Figure 3). The coordinate origin O of the carrier coordinate system is located at the center of gravity of the carrier, with Xb pointing to the right side of the carrier, Yb pointing to the longitudinal axis of the carrier, and Zb pointing to the vertical axis of the carrier. The orientation of the carrier coordinate system relative to the geographic coordinate system is the attitude angle of the carrier.

Figure 3 Carrier coordinate system

5. Navigation coordinate system (n-system) - OXnYnZn

The navigation coordinate system is a coordinate system selected as the navigation reference based on the needs of the navigation system during navigation. When the navigation coordinate system is selected to coincide with the geographic coordinate system, this navigation coordinate system can be referred to as the north pointing system; In order to meet the needs of navigation near the polar regions, the Zn axis of the navigation coordinate system is often selected to coincide with the Zt axis, resulting in a difference of one free azimuth or swimming azimuth between Xn and Xt and Yn and Yt. This type of navigation coordinate system can be called a free azimuth system or swimming free azimuth system. This article uses the geographic coordinate system as the navigation coordinate system.

6. Platform coordinate system (p-system) - OXpYpZp

The platform coordinate system is the coordinate system obtained when reproducing the navigation coordinate system using an inertial navigation system. The coordinate origin O of the platform coordinate system is located at the center of gravity of the carrier. When there is no error in the inertial navigation system, the platform coordinate system coincides with the navigation coordinate system; When there is an error in the inertial navigation system, the platform coordinate system will have an error angle relative to the navigation coordinate system. For platform based inertial navigation systems, the platform coordinate system is implemented through the platform body; For the strapdown inertial navigation system, the platform coordinate system is implemented through the attitude matrix stored in the computer, hence it is also called a "mathematical platform".

Working principle of strapdown inertial navigation system

Figure 4 Strapdown Inertial Navigation System Block Diagram

In order to overturn the mathematical model of the strapdown system, we define the following parameters:

1. Location

L: Local latitude, λ: local latitude, h: local altitude

2. Attitude angle

ψ: The heading angle of the carrier. The angle between the projection of the longitudinal axis of the carrier on the horizontal plane and the geographic meridian is called the heading angle. The value of heading angle is calculated counterclockwise starting from the geographic north direction.

γ: The roll angle (also known as the tilt angle) of the carrier. The angle between the longitudinal symmetry plane of the carrier and the longitudinal vertical plane

It is the roll angle. The roll angle is calculated from the vertical plane, with right inclination being positive and left inclination being negative.

θ: The pitch angle of the carrier. The angle between the longitudinal axis of the carrier and the longitudinal horizontal axis is the pitch angle, which is positive upwards and negative downwards.

3. Bi Li

Specific force refers to the algebraic sum of displacement acceleration and gravitational acceleration per unit mass, that is, the algebraic sum of external forces acting on a unit mass. Fn: specific force in the navigation coordinate system, fb: specific force in the carrier coordinate system.

4. Angular velocity

Angular velocity is represented by symbols with subscripts, such as ω bibx, where the subscript represents the rotational angular velocity of the b system (carrier coordinate system) relative to the i system (inertial coordinate system), the superscript represents the projection of this angular velocity in the b system (carrier coordinate system), and x represents the component projected on the x-axis. Other angular velocity symbols have similar meanings.

5. Speed

Velocity is also represented by symbols with subscripts, such as Vtetx, where the subscript represents the velocity of the t system (geographic coordinate system) relative to the e system (Earth coordinate system), the superscript represents the projection of this velocity in the t system (geographic coordinate system), and x represents the component projected on the x-axis.

6. Coordinate system transformation matrix

The coordinate system transformation matrix is also represented by symbols with subscripts, such as Cnb, which means the transformation matrix from the b system (carrier coordinate system) to the n system (navigation coordinate system). The meaning of the symbols for other coordinate system transformation matrices is similar.

7. Earth's radius

If we consider the Earth as an ellipsoid, the equatorial radius Re of the Earth is 6378393m, and the ellipticity e is 1/298.257.

8. Earth's rotational angular velocity

ω ie=15.0411 °/hr=7.29212 * 105 rad/s.

9. Gravitational acceleration

The formula for gravitational acceleration can be approximated by the following equation: g=g0 * (1+0.005271 * sin2L) -3.086 * 10-6h, where g0 is the value of gravitational acceleration on the equatorial surface, g0=9.7803267714 m/s2。

The mathematical model of the strapdown inertial navigation system mainly includes the navigation position equation, velocity equation, and attitude equation.

Figure 5 Schematic diagram of navigation calculation for strapdown system

The calculation formulas for each module are as follows:

1. Instant correction of quaternion Q

There are three commonly used algorithms for real-time correction of strapdown matrices, namely Euler angle method, direction cosine method, and quaternion method. Due to the fact that the strapdown matrix is obtained through computer calculations, there is an issue of orthogonalization in the calculation of the strapdown matrix. The criteria for evaluating the advantages and disadvantages of algorithms should be summarized as follows: under the same total computational load, after orthogonalization, the algorithm research and simulation of the strapdown inertial navigation system have the smallest algorithm error. The best method is the quaternion algorithm, which is currently widely used in real-time correction of strapdown inertial navigation systems.

Choose quaternion method as the real-time correction algorithm for the strapdown matrix. Quaternion refers to a number consisting of one real unit 1 and three imaginary units, and having the following real elements:

The quaternion of rotation of the carrier coordinate system relative to the platform coordinate system is:

The immediate correction of Q can be achieved by solving the following quaternion differential equation:

When solving the above formula, the initial values of quaternions are required. The initial value of quaternions can be determined based on the elements in the initial value of the attitude matrix determined in the initial alignment, and by utilizing the equal relationship between the corresponding elements of quaternions and attitude matrices.

2. Calculation of Strapdown Matrix T

After obtaining the quaternion from equation (1), the strap down matrix can be calculated based on equation (4).

3. Optimal Normalization of Quaternion Q

Due to algorithm errors in computers, the strapdown matrix can become a non orthogonal matrix. Orthogonal processing of the strapdown matrix can eliminate the influence of non orthogonal algorithm error sources. Normalize quaternions, which means completing the orthogonalization of the strap down matrix Ding. The optimal normalization of quaternions based on the minimum Euclidean norm can be obtained by the following equation:

4. Coordinate Conversion of Specific Forces

The specific force measured by the accelerometer can be converted into matrix T, that is:

5. Instant correction of speed

The basic equation of inertial navigation is:

Write it in matrix form:

6. Position velocity calculation

Among them:

The latitude L in the formula is obtained through real-time correction of latitude, where Re is the equatorial radius of the Earth and e is the ellipticity of the Earth.

7. Calculation of Earth's velocity

Conclusion

Strapdown inertial navigation is currently the main direction of navigation technology development. The use of modern simulation technology to develop SINS simulation systems that are very similar to real systems in time and space greatly simplifies the analysis and research of various attitude algorithms and alignment schemes. It plays an important role in optimizing the design of Jielian inertial navigation systems, saving development costs, and accelerating development progress.

Due to the maturity of accelerometer technology, fiber optic gyroscope is the most important sensing component in the strapdown inertial navigation system, and its performance directly affects the overall performance of the strapdown system. The AgileLight series fiber optic gyroscope from Huilian adopts IntelliProcess technology, with low zero bias and random walk, making it the best choice for strapdown inertial navigation systems.