Inertial Navigation

Abstract

There is a story in “The Adventures of Sherlock Holmes”: Holmes is kidnapped and blindfolded, tied up in a carriage. Holmes estimates the time spent by silently counting the time, and then calculates the distance based on the speed of the carriage to determine the approximate location of the kidnapper. The method used by Sherlock Holmes is Dead Reckoning (DR).

There is a story in “The Adventures of Sherlock Holmes”: Holmes is kidnapped and blindfolded, tied up in a carriage. Holmes estimates the time spent by silently counting the time, and then calculates the distance based on the speed of the carriage to determine the approximate location of the kidnapper. The method used by Sherlock Holmes is Dead Reckoning (DR).

There is a physiological organ in human body that adopts a similar principle of DR. The vestibular organ of the inner ear is the balance receptor organ in human body, which can sense the changes of horizontal or vertical linear acceleration. When our vehicle rotates or turns, such as when the car turns and the aircraft makes circular motion, the angular acceleration acts on the corresponding semicircular canals on both sides of the inner ear. When hair cell in the ampulla of one side of the semicircular canals are stimulated to bend and deform to produce a positive potential, the opposite hair cell will bend and deform to produce a negative potential. Similarly, when the linear acceleration (deceleration) changes of the riding tools, such as car starting, deceleration braking, ship shaking, bumping, elevator and aircraft ascending or descending, these stimuli cause the deformation discharge of the hair cell in the capsule spots of the vestibular elliptical bursa and saccule to transmit and sense the operation information to the central nervous system. So, the eyes help us directly locate and navigate based on external information, while ears help us determine attitude and infer orientation.

Dead reckoning (DR) is the primary form of this approach, and inertial navigation is the advanced form.

In navigation technology, dead reckoning is particularly important as it does not require external information to realize navigation. Especially in many conditions where external information cannot be obtained, such as submarine navigation, night driving or external information is often disturbed, direct positioning methods cannot be used any more, but the navigation based on DR has become the only autonomous method that can work normally. It calculates the current or future position from past known positions based on the measurement of parameters such as the carrier’s heading, speed, and time, thereby obtaining a motion trajectory. Early days of sailing, the heading was determined by magnetic compass or compass, the velocity was determined by log, time was displayed in the chronometer, and the ship position was calculated by using these parameters. The INS widely used in modern times uses the accelerometer to measure the acceleration of the carrier, and by integrating the acceleration twice, the calculated position, velocity and attitude of the carrier can be continuously output.

8.1 Principle of Dead Reckoning

8.1.1 Brief Introduction of Dead Reckoning

Dead reckoning obtains the current position by measuring position changes or measuring velocity and integrating it with the original position (Fig. 8.1). Only by projecting the velocity and displacement on the carrier coordinate system to the reference coordinate through the attitude parameters can the velocity or range of the carrier relative to the reference coordinate system be calculated. For 2D navigation application, heading is sufficient. For 3D navigation application, it is necessary to measure three components of attitude parameters. When the attitude changes, the smaller the step size for position calculation adopt, the more accurate navigation parameters can be obtained. In the past, it was mostly done manually, which severely limited the data update rate. Now, it is done by computer.

Fig. 8.1

Principle of dead reckoning

8.1.2 Principle of Dead Reckoning

8.1.2.1 Dead Reckoning of Ship

Dead reckoning is the basic method to obtain the position of the ship at any time in the voyage [24]. Under the condition of not relying on external navigation targets, relying solely on the most basic heading and speed indication equipment of the carrier, as well as external wind and current data, starting from the known starting point of the calculation, a certain precision speed and carrier position at a certain time can be calculated. Through trajectory calculation, the pilot can clearly understand the continuous trajectory of the ship’s motion at sea. In modern ships, the function of manually plotting and automatically inferring the flight path is completed by a flight path plotter (see Chap. 10) or a navigation workstation.

• (1) Classification

There are two kinds of dead reckoning. The first is track plotting, which is used to solve the two problems. One is plotting the ship trajectory according to heading, speed, sailing distance, wind and current. The other is plotting the ship heading and obtaining the estimated position according to the planned route, wind and current. This method is simple and intuitive, and it is the main method for the pilot to calculate ship position during sailing.

The second is track calculating, which uses mathematical equations to calculate the ship position and then plots the ship position on the chart according to heading, speed, sailing distance, wind and current. This method is the theoretical basis of computer track calculation and ship steering automation.

• (2) Basic method

Given the compass heading, log range, and wind and current elements, calculate the trajectory direction. The effects of wind and current on ship navigation often coexist. The comprehensive impact of wind and current on ship navigation is discussed based on a separate analysis of the effects of wind and current. Due to the different characteristics of the influence of wind and flow on ship navigation, when analyzing the combined effects of wind and flow, the influence of wind is generally analyzed first, and then the influence of flow, that is, the drawing method of “wind before flow” is adopted.

As shown in Fig. 8.2, when the ship operates with speed V_E and heading C_T, under the influence of the wind, the ship deviates from a leeway angle α and moves along the route line in the wind at speed V_α. As a result of the current, the ship deviates from a drift angle β. A drift triangle is made on the route line in the wind to obtain the final velocity V_γ and track in the wind and current.

Fig. 8.2

Wind and current pressure difference

The true course and speed should be replaced by the track in wind and speed in wind when the wind and current simultaneously affect the drift triangle. The six elements of the drift triangle are track in wind, speed in wind, current direction, current velocity, track in wind and current, and speed in wind and current. Speed V_E is usually used instead of wind speed V_α when the wind speed is unknown.

To calculate a ship’s position by sailing in the wind or current, the range can be calculated based on the range S_L or wind speed recorded (V_α ×  Δt) by the log. A point can be measured on the wind track line, and then a flow parallel line intersects with the wind track line from that point. The intersection point is the calculated ship’s position at that time. The range in the wind can also be calculated based on the obtained speed and sailing time in the wind, and the ship position can be directly calculated by intercepting it on the track line in the wind.

Finally, it should be noted that under the combined influence of wind and current, although the ship moves along the track line in the wind and current, the direction indicated by the bow is still the direction of the original heading line.

8.1.2.2 Dead Reckoning of Pedestrian

DR of pedestrian is one of the most challenging applications in navigation technology. The pedestrian navigation system must be able to operate normally in urban areas, under trees, and even indoors where GNSS and most other radio navigation systems have poor performance. Inertial sensors can be used to measure forward motion through dead reckoning. However, pedestrian applications generally require small size, light weight, low power consumption, and in most cases, low cost, therefore MEMS sensors are often used. When used MEMS alone, the performance of MEMS is poor, resulting in limited accuracy of calculation. Low dynamic and high vibration environmental conditions also limit the effectiveness of GNSS or other positioning systems in correcting effect.

8.1.3 Characteristics of Dead Reckoning

Because dead reckoning requires the use of previously known positions, heading, and estimated speed to derive the current position over time, the new position calculated is only derived from the previous step value, and the error and uncertainty of the derived position estimation value increase with time. Therefore, it is often used in short time or low precision situations.

8.2 Principle of Inertial Navigation

Inertial navigation uses inertial sensors (gyroscopes, accelerometers) to measure the linear and angular motion of the carrier relative to the inertial space. When the initial conditions are given, the output of the carrier’s heading, attitude, speed, position and other navigation information can be provided. INS (INS) is a powerful navigation equipment with the advantages of autonomy, concealment and so on. It is the most important mean of navigation and positioning of underwater vehicles and plays an irreplaceable role in military applications.

8.2.1 Basic Principles of Inertial Navigation

8.2.1.1 Citation

Let the carrier move uniformly in a straight line, the motion time be t, the velocity be V, the acceleration be a and the travel distance be S, the relationship between the above parameters is as follows:

$$ \begin{aligned} V & = V_{0} + at \\ S & = S_{0} + V_{0} t + \frac{1}{2}at^{2} \\ \end{aligned} $$

(8.1)

When all the initial condition is zero, i.e., $S_{0} = V_{0} = 0$ ($S_{0}$ is the initial distance and $V_{0}$ is the initial velocity), then:

$$ \begin{aligned} V & = at \\ S & = \frac{1}{2}at^{2} \\ \end{aligned} $$

(8.2)

Because the carrier’s heading and speed change at time, the acceleration a will not be a constant and can no longer be solved by algebraic operation at this time but need to continuously measure the acceleration a and perform recursive calculation. The recursive calculation process is as follows:

• (1) The acceleration is integrated once to obtain the velocity of the carrier (velocity update):

$$ \left\{ \begin{gathered} V_{E} \left( {k + 1} \right) = V_{E} \left( k \right) + \int\limits_{0}^{\tau } {a_{E} \left( k \right)dt} \hfill \\ V_{N} \left( {k + 1} \right) = V_{N} \left( k \right) + \int\limits_{0}^{\tau } {a_{N} \left( k \right)dt} \hfill \\ \end{gathered} \right. $$

(8.3)

where V_E(k + 1) and V_N(k + 1) are the No. k + 1 recursion values of the eastward and northward velocities of the vehicle. V_E(k) and V_N(k) are the No. k recursion values of the eastward and northward velocities of the vehicle. a_E(k) and a_N(k) are the No. k recursion values of the eastward and northward accelerations of the vehicle. τ is the sampling time interval.

• (2) The velocity is integrated to obtain the range of the carrier (position update):

$$ \left\{ {\begin{array}{*{20}l} {\varphi \left( {k + 1} \right) = \varphi \left( k \right) + \frac{1}{R}\int\limits_{0}^{\tau } {V_{N} \left( k \right)dt} } \\ {\lambda \left( {k + 1} \right) = \lambda \left( k \right) + \frac{1}{{R\cos \left[ {\varphi \left( k \right)} \right]}}\int\limits_{0}^{\tau } {V_{E} \left( k \right)dt} } \\ \end{array} } \right. $$

(8.4)

where φ(k + 1) and λ(k + 1) are the No. k + 1 recurrence values of the latitude and longitude of the vehicle. φ(k) and λ(k) are the No. k recurrence values of the latitude and longitude of the vehicle.

This is the basic principle of the horizontal north pointing semi-analytic INS.

8.2.1.2 Platform Coordinate System and Attitude Determination

The difficulty to realize inertial navigation is how to keep the measuring direction of the accelerometer always along the east and the north under the condition of carrier maneuver, although the basic principle of inertial navigation is very simple. The key to solving this problem is to build the three-axis inertial stabilization platform. When the platform tracks the local geographical coordinate system stably, the three axes of the platform always point to the east, north and zenith. In this way, the carrier attitude can be obtained and output directly by the angle measuring elements installed on the gimbal of the platform, and the sensitive axis of the accelerometer A_E and A_N can always point to the east‒west or north‒south direction respectively even in maneuver process. Once the platform coordinate system cannot be guaranteed to be consistent with the geographical coordinate system, it will bring errors to the accelerometer measurement and the carrier attitude angle measurement. So it is very important to establish an accurate platform coordinate system.

How to ensure platform level and north pointing, that is, how to keep stable platform three-axis tracking geographic coordinate system. This issue is described in Chap. 7. It is assumed that the stable platform has achieved the alignment of the geographical coordinate system according to the working mode of the stabilized gyrocompass introduced in Chap. 7. Taking the moving base as an example, when the carrier P moves on the Earth surface (Fig. 8.3), the rotation angular velocity of the geographical coordinate system ENT with P as the origin relative to the inertial space will consist of two parts: one is the rotation angular velocity of the Earth ω_ie, that is, the rotation angular velocity of the Earth relative to the inertial space; the other is the rotation angular velocity ω_en caused by the movement of the carrier relative to the Earth. By controlling the precession of the gyroscopes on the stabilized platform according to the magnitude of the three-axis directional component of the angular velocity of rotation in the geographic coordinate system, stable tracking of the stabilized platform can be achieved.

Fig. 8.3

Rotational angular velocity of the local geographic coordinate system caused by vehicle motion

The component expressions of rotation angle velocity in the local geographical coordinate system caused by carrier motion are as follows:

$$ \begin{aligned} \omega_{E} & = - \frac{{V_{N} }}{R} = \frac{{V_{IN} }}{R} \\ \omega_{N} & = \frac{{V_{E} }}{R} + \omega_{e} \cos \varphi = \frac{{V_{IE} }}{R} \\ \omega_{T} & = \frac{{V_{E} }}{R}\tan \varphi + \omega_{e} \sin \varphi = \frac{{V_{IE} }}{R}\tan \varphi \\ \end{aligned} $$

(8.5)

where:

V_E, V_N: the eastward and northward component of the linear velocity of the carrier relative to the geographical coordinate system.

V_IE, V_IN: the eastward and northward component of the linear velocity of the carrier relative to the inertial space (geocentric inertial coordinate system).

The attitude of the carrier can be measured by the angle sensors installed on the three axes of the stable platform when it has tracked the local geographical coordinate system stably.

8.2.1.3 Calculation of Velocity

Eastern and northward accelerometers installed on the stabilization platform are used to measure the acceleration of the vehicle in the east‒west direction and north‒south direction. However, in fact, according to the equivalent effect of gravitation and acceleration in general relativity theory, the accelerometer measurement will include inertial force and gravitation, which is difficult to distinguish from each other. The combination of them measured by the accelerometer is called the specific force.

$$ {\text{Specific}}\,{\text{force}} = {\text{inertial}}\,{\text{force}}{-}{\text{gravitation}} $$

(8.6)

The gimballed inertial navigation system adopts stable platform to keep the accelerometer in a horizontal state, thereby achieving isolation of attraction containing gravity. Once there is an error in the platform level, it will lead to errors in acceleration measurement. When calculating carrier velocity based on measured acceleration, it is necessary to fully consider the navigation coordinate system where the velocity located. Once the navigation coordinate system is a non-inertial coordinate system, the motion influence of the navigation coordinate system must be compensated. The harmful acceleration caused by the earth rotation and ship motion in the acceleration output information needs to be compensated by computer, so that the accurate horizontal acceleration can be obtained. The east and north speed components of the ship can be obtained by computer integration.

According to Newton’s mechanics principle, when the reference coordinate system (e.g., the earth) rotates at a constant angular rate:

$$ \begin{aligned} {\text{Inertial}}\,{\text{acceleration}} & = {\text{relative}}\,{\text{acceleration}} + {\text{implicated}}\,{\text{acceleration}} \\ & \quad + {\text{Coriolis}}\,{\text{acceleration}} \\ \end{aligned} $$

(8.7)

Therefore, by introducing Formula (8.7) into Formula (8.6), we can obtain (8.8):

$$ \begin{aligned} {\text{Specific}}\,{\text{force}} & = {\text{relative}}\,{\text{acceleration}}\,{\text{force}} + {\text{Coriolis}}\,{\text{acceleration}}\,{\text{force }} \\ & \quad {\text{ + implicated}}\,{\text{acceleration}}\,{\text{force}}{-}{\text{gravitation}} \\ \end{aligned} $$

(8.8)

On Earth, gravity is the combination of centrifugal acceleration and gravitation. In summary, if the specific force measured by the accelerometer is $\overline{\varvec{f}}$, the relative acceleration of the vehicle relative to the reference frame is $\dot{\overline{\varvec{V}}}_{r}$, the angular velocity of the earth’s rotation is $\overline{\omega }_{ie}$, the angular velocity of the rotation of the vehicle relative to the earth is $\overline{\omega }_{eb}$, and the local gravity is $\overline{\varvec{g}}$.

$$ \frac{{\overline{\varvec{f}}}}{m} = \dot{\overline{\varvec{V}}}_{r} + \left( {2\overline{\omega }_{ie} + \overline{\omega }_{eb} } \right) \times \overline{\varvec{V}}_{r} - \overline{\varvec{g}} $$

(8.9)

The above formula is expanded into three-dimensional component form, and the east and north accelerations of vehicles a_E and a_N are assumed to be:

$$ \begin{aligned} f_{E} & = a_{E} - \left( {2\omega_{ie} \sin \varphi + \frac{{V_{E} }}{R}\tan \varphi } \right)V_{N} + \left( {2\omega_{ie} \cos \varphi + \frac{{V_{E} }}{R}} \right)V_{\varsigma} \\ f_{N} & = a_{N} + \left( {2\omega_{ie} \sin \varphi + \frac{{V_{E} }}{R}\tan \varphi } \right)V_{E} + \frac{{V_{N} }}{R}V_{\varsigma} \\ f_{\varsigma} & = a_{\varsigma } - \left( {2\omega_{ie} \cos \varphi + \frac{{V_{E} }}{R}} \right)V_{E} - \frac{{V_{N} }}{R}V_{N} + g \\ \end{aligned} $$

(8.10)

For ships and vehicles moving on the Earth surface, $V_{\zeta } \approx 0$. Therefore, it can be obtained from the above:

$$ \begin{aligned} a_{E} & = f_{E} + \left( {2\omega_{ie} \sin \varphi + \frac{{V_{E} }}{R}\tan \varphi } \right)V_{N} \\ a_{N} & = f_{N} - \left( {2\omega_{ie} \sin \varphi + \frac{{V_{E} }}{{R_{N} }}\tan \varphi } \right)V_{E} \\ \end{aligned} $$

(8.11)

According to Formula (8.11), a_E and a_N are obtained. Then, the eastward and northward velocities V_E and V_N at time k can be calculated according to (8.4).

8.2.1.4 Calculation of Longitude and Latitude

The velocities V_E and V_N are integrated to obtain the displacement of the carrier. When using longitude and latitude as the positioning parameters, the displacement is added with the initial position λ₀, φ₀ to obtain the longitude and latitude λ, φ of the carrier. When the Earth is regraded as sphere whose radius is R, the longitude and latitude λ and φ of the carrier can be obtained from the following Formula (8.12):

$$ \begin{aligned} \varphi \left( t \right) & = \frac{1}{R}\int\limits_{0}^{t} {V_{N} dt + \varphi_{0} } \\ \lambda \left( t \right) & = \frac{1}{R}\int\limits_{0}^{t} {V_{E} \frac{1}{\cos \varphi }dt + \lambda_{0} } \end{aligned} $$

(8.12)

where λ₀ and φ₀ are the initial longitude and latitude of the vehicle.

The formulas above are the basic principles of INSs. The block diagram of the horizontal north pointing semi-analytic INS is drawn in Fig. 8.4. The INS is based on Newton mechanics [25]. Linear motion and angular motion relative to inertial space measured by accelerometer and gyroscope. Therefore, accelerometers and gyroscopes are collectively called “inertial sensors”.

Fig. 8.4

Block diagram of horizontal north pointing semi-analytic INS

8.2.2 Gimballed Inertial Navigation

8.2.2.1 Basic Structure of the Gimballed Inertial Navigation System

According to whether the inertial stabilization platform is adopted or not, inertial navigation systems can be divided into two categories: Gimballed Inertial Navigation System (GINS) and Strapdown Inertial Navigation System (SINS). SINS is such a system in which inertial sensors are directly installed on the vehicle, in which the concept of the navigation platform is replaced by the “mathematical platform” established by computers.

Different types of INSs have different components. Even for the same type of INSs installed on different carriers, their composition will be different. In most cases, the INS consists of the following functional components:

• (1) Main instrument: This is also known as the inertial platform, which is the core of the INS. It consists of an inertial platform, shock absorber, temperature control system and electrical components.

• (2) Navigation computer: This is mainly used for calculating navigation parameters and command signals applied to gyro torquers. Digital computers are important component of INSs, which accomplishes all calculations and provides control information and data output.

• (3) Control and display device: The console is used to operate and control the INS. It includes a working state switch, initial data setting, output data display and fault alarm indication.

• (4) Power supply device: power supply is used to supply AC/DC power for electrical components, inertial components and various loops in INSs with high performance requirements.

• (5) Electronic equipment cabinet: This includes stable circuit amplifier, accelerometer circuit amplifier, start-up circuit amplifier, inertial components and temperature control circuit of the platform.

• (6) Signal transmitter and peripheral equipment: This part is different according to the different requirement of INS applied to various vehicles. Marine INS are often equipped with heading, pitch and rolling transmitter systems.

8.2.2.2 Classification of Gimballed Inertial Navigation

According to the platform coordinate system established by the stabilized platform, GINS can be divided into three kinds:

• (1) Analytic GINS

The analytic GINS has three-axis gyro stabilized platform, which is stable in inertial space (system i), so it is also called space stabilized INS.

Three mutually perpendicular gyroscopes and accelerometers are installed on the stabilized platform. The accelerometer measures acceleration of the vehicle in the inertial coordinate system and the gravitation component. When calculating the velocity and position of the vehicle in the inertial coordinate system, the acceleration of the vehicle can be obtained by calculating and eliminating the gravitation influence without modifying the influence of the Earth rotation and the motion of the vehicle. For vehicles moving near the Earth surface, when calculating the velocity and position of the vehicle in the Earth coordinate system or geographic coordinate system, the velocity and position of the vehicle in the inertial coordinate system must be transformed to obtain the velocity, longitude and latitude relative to the Earth coordinate system or geographic coordinate system. Compared with the local horizontal INS, the spatial orientation taken by the platform cannot separate the acceleration of motion from gravity. The data measured by the accelerometer must be analyzed and calculated by computer to obtain the velocity and position of the vehicle, so it is also called the analytic INS.

The platform structure of analytic GINS can be simplified, but due to the need to solve the problems of gravity correction and coordinate conversion, the calculation is heavy. Analytic GINS is applicable to intercontinental missiles, space probes and other carriers flying far away from the Earth.

• (2) Semi-analytical GINS

The two horizontal axes of the three-axis stabilized platform of the semi-analytical INS are always in the local horizontal plane, and the vertical axes coincide with the ground vertical. The azimuth refers to the north or a certain direction. This type is also known as the local horizontal INS. Semi analytical INS have the following types.

• a. North-pointing INS

The inertial stabilization platform of the north-pointing INS tracks and stabilizes in the local geographic coordinate system, that is, the platform horizontally points to north. It is also called semi-analytical north-pointing INS. North-pointing INS is the most common GINS and suitable for aircraft, ship, combat vehicle and other vehicle moving near the earth surface.

• b. Free azimuth INS

The platform of the free azimuth INS is stable in the horizontal plane, while the azimuth axis is not controlled and stable in the inertial space.

• c. Wandering azimuth INS

The platform of the wandering azimuth INS is stable in the horizontal plane, and the azimuth axis is controlled by the earth rotation angle speed $\user2{\omega }_{{{\text{ie}}}}$, so that the platform rotates around the azimuth axis in space with $\user2{\omega }_{{{\text{ie}}}}$. If on the stationary base, the north- pointing INS and the wandering azimuth INS are the same.

In the semi-analytical GINS, gyroscopes and accelerometers are mounted on stable platform. Therefore, the acceleration measured by the accelerometer is the horizontal and vertical component of the carrier relative to inertial space. Because the platform keeps horizontal, the accelerometer output signal does not contain the component of gravity g but contains the harmful acceleration caused by the rotation of Earth and the movement of vehicle. Therefore, the velocity and position of the carrier relative to the earth can only be calculated by integration after the harmful acceleration caused by Earth’s rotation and carrier speed is eliminated. Since the vertical acceleration of ships and other vehicles is usually small, the vertical channel accelerometer can be omitted to simplify the calculation of harmful acceleration and system calculation. The most commonly used navigation coordinate system is the local horizontal coordinate system, especially the local geographic coordinate system, because the longitude and latitude calculation in this coordinate system is the most direct and simplest.

• (3) Geometric GINS

There are two platforms in this type of INS: one is used to install gyroscopes, which is stable in inertial space; the other is used to install accelerometers, which is stable in the geographic coordinate system (that is, two horizontal axes, one pointing east, one pointing north, and they are always in the horizontal plane). The rotation axis between the two platforms is rotated at the angular velocity of the Earth rotation. At the starting point, the direction of the rotation axis should be adjusted to be parallel to that of the Earth self-rotation axis. Because the platform with the accelerometers tracks the gravity direction and the platform with the gyroscopes is stable in inertial space, this system is also called the gravity INS. According to the geometric relationship between the two platforms of the system, the longitude and latitude of the vehicle can be determined. Therefore, it is called a geometric INS. It has high precision, can work for long time, and has a small amount of calculation, but the structure of the platform is complex. It is mainly used for navigation and positioning of submarines.

8.2.2.3 Basic Characteristics of Inertial Navigation

Because the INS calculates the acceleration of the vehicle relative to the geographic coordinate system by measuring and calculating the acceleration, the linear velocity is obtained by integrating the acceleration once, and the position information is obtained by integrating the acceleration twice [26]. Therefore, INS is actually a dead reckoning system. Its main characteristics are as follows:

• (1) Capable of working independently and continuously for long time. INS can work independently without receiving any external information. It has good concealment and is not affected by geographical, meteorological and other external environment and time constraints. It can be widely used in many fields, such as aviation, aerospace, marine and land navigation. Especially in underwater navigation, INS is the most important navigation equipment.

• (2) Providing complete vehicle navigation information. It can provide vehicle navigation parameters such as heading, pitch, roll, east speed, north speed, longitude and latitude with good continuity and low noise.

• (3) The positioning error accumulates and diverges with time. The main reason is the divergence of the longitude error and the periodicity of the latitude error. The overall effect is that the positioning error of INS will increase with time. Therefore, the positioning accuracy of INS must specify the time range, for example 1 nautical mile/24 h. For applications where the working time exceeds the designed time duration, the errors need to be corrected periodically, which is called inertial navigation readjustment, to maintain long-term positioning performance.

• (4) The errors of output parameters, such as heading, attitude and velocity, are periodic. It mainly includes 84.4 min of Schuler periodic oscillation, Foucault periodic oscillation related to geographic latitude and 24 h of Earth periodic oscillation. When applied to systems with different working time durations, the error characteristics of INS are different. For example, when used in short-time applications, such as air-to-air missiles, because the working time is much lower than the Schuler period, the error reflects the characteristics of approximately linear growth. When applied to submarines that work for several days, all the oscillation errors of the INS will appear. Because of the application of random errors of inertial components, system errors (such as course errors) will also diverge, and the velocity damping technology is needed to suppress them.

• (5) INS has long start-up time. To reduce the initial alignment error of the platform, the system adopts self-alignment methods such as coarse alignment and precise alignment, so the start-up time is long. In the high-precision application field, the start-up process also needs to wait for the gyroscope error characteristics to be stable and can be accurately calibrated, the start-up time is always more than 1 day. For some time-demanding applications, the transfer alignment method is often used to achieve rapid start-up by using higher precision inertial navigation (main INS) speed and attitude data.

• (6) The precision of the INS mainly depends on the precision of the gyroscope and accelerometer, the errors of the initial position and velocity, and the initial alignment error of the platform. The performance of the gyroscope determines the positioning accuracy of the INS.

• (7) There are differences in the characteristics of inertial technology in different application fields. The characteristics of the aerospace field are high reliability, high accuracy, long endurance, and small size. The marine field emphasizes high reliability, high precision, and long endurance. The aviation industry emphasizes fast start-up, small size, high maneuverability, and strong environmental adaptability. The field of land and petroleum geological exploration and measurement emphasizes fast start-up, small size, and good environmental adaptability. The field of robotics requires fast start-up, small size, light weight, and so on.

8.3 Principle of Strapdown Inertial Navigation

The mature application of solid-state gyroscopes such as ring laser gyroscopes (RLGs), fiber optic gyroscopes (FOGs) and microelectromechanical system (MEMS) gyroscopes has accelerated the development of strapdown inertial navigation technology. The strapdown inertial navigation system (SINS) has developed rapidly in the fields of aeronautics and aerospace since the mid-1970s. They have been widely used and are gradually replacing the GINS.

8.3.1 Brief Introduction of SINS

SINS and GINS are basically the same in configuration. They are mainly composed of Inertial Measuring Unit (IMU), navigation computer and display device. Figure 8.5 shows the block diagram of the SINS. The combination of gyroscopes and accelerometers is commonly called inertial combination. The pointing of the three input axes of gyroscopes and accelerometers should be strictly orthogonal, and the IMU should be installed directly on the vehicle in full accordance with the body coordinate system of the vehicle.

Fig. 8.5

Block diagram of the SINS

SINSs have no stable platform but directly connect inertial combination to the vehicle, so attitude and heading cannot be directly measured by IMU. To obtain the heading and attitude, the output of the gyroscope and accelerometer must be processed in the navigation computer. The attitude matrix of the body coordinate system relative to the geographic coordinate system can be obtained by calculation, and then a ‘mathematical platform’ in the computer can be established with the heading and attitude calculated. With the help of the ‘mathematical platform’, the speed and position of the vehicle can be calculated.

SINS can provide position, speed and attitude information, while the strapdown Attitude and Heading Reference System (AHRS) can only provide heading and attitude information. Although there are differences in functions between SINS and AHRS, their basic structures are the same, and their principles are similar and closely connected.

Compared with the stabilized gyrocompass (Chap. 7.3), the AHRS requires high real-time and continuity. Generally, shock absorber is always not installed to improve the response frequency. It has short response time and high frequency of data output. Therefore, AHRS requires high bandwidth, minor data delay and high accuracy. In different applications, AHRSs can be configured differently, but the different requirements of the jitter effect, initial alignment, quasi-instantaneous start-up and structural resonance should be considered (Fig. 8.6).

Fig. 8.6

Nautical -IIG electric compass, with permission from [CSSC Marine Technology]

At present, many companies around the world produce RLG gyrocompass and FOG gyrocompass, such as the United States Kearfott, Honeywell, Northrop Grumman, L-3 Communications company, French Thales, Sagem, IXSea company, German iMAR company, and Russian Astrophysika company (Fig. 8.7).

Fig. 8.7

Conch-90 INS, with permission from [CSSC Marine Technology]

8.3.2 Principle of Strapdown Inertial Navigation

8.3.2.1 The Solution Process of SINS

Unlike GINS, there is no stable platform in SINS, so the specific force measured by the accelerometer must be projected to the navigation coordinate system through the attitude matrix firstly to complete the subsequent acceleration calculation, velocity calculation and position calculation. For ease of understanding, discuss the following example [27].

For the convenience of the following discussion, it is assumed that the car with a navigation system is limited to a single plane, and its schematic block diagram is shown in Fig. 8.8.

Fig. 8.8

Principle block diagram of two-dimensional SINS

Two accelerometers and a uniaxial rate gyroscope are rigidly mounted on the car base. The sensitive axes of the accelerometer (Fig. 8.9) are perpendicular to each other and are consistent with the axes of the car in the motion plane, represented as x_b and z_b. The sensitive axis of the gyroscope (y_b) is installed perpendicular to the accelerometer two sensitive axes, which measure rotation around the axis perpendicular to the motion plane. Assuming navigation in the reference coordinate system represented by x_i and z_i, the relationship between the vehicle body coordinate system and reference coordinate system is shown in Fig. 8.9. In the figure, θ represents the angular displacement between the reference coordinate system and the body coordinate system. Attitude θ can be determined from the integral of the angular rate ω_yb measurable by the gyroscope, as shown in Fig. 8.8:

$$ \dot{\theta } = \omega_{yb} $$

(8.13)

Fig. 8.9

Reference coordinate system of two-dimensional

After obtaining θ, the specific force of accelerometer-measured f_xb and f_zb can be projected to the reference coordinate system OX_iZ_i.

$$ \left[ \begin{gathered} f_{xi} \hfill \\ f_{zi} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}l} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]\left[ \begin{gathered} f_{xb} \hfill \\ f_{zb} \hfill \\ \end{gathered} \right] $$

(8.14)

The attitude conversion matrix in the above formula is the key to the strapdown system, which is used to project the specific force of the accelerometer from its sensitive axes to the reference coordinate system axes. This is different from the gimbaled system. Similar to GINS, the relative acceleration of the object relative to the reference frame can be calculated by Formula (8.9) through specific forces f_xi and f_zi. The relative velocity and position can be obtained by integrating the relative acceleration twice.

In summary, the SINS solution-solving process is summarized as follows:

• (1) Measure the angular velocity of the vehicle relative to the inertial space by the gyroscope and calculate the attitude matrix;

• (2) Project the specific force measured by the accelerometer to the reference coordinate system by the attitude matrix;

• (3) Analyze vehicle motion and calculate relative acceleration;

• (4) Integrate the relative acceleration to obtain the velocity update;

• (5) Integrate the relative velocity to obtain the position update.

8.3.2.2 Attitude Update

The digital platform of SINS uses the angular velocities measured by strapdown gyroscopes to calculate the attitude matrix, extracts the attitude and heading information of the vehicle from the elements of the attitude matrix, uses the attitude matrix to transform the output of the accelerometer from the body coordinate system to the navigation coordinate system, and then carries out navigation calculations. There are four main methods for describing the attitude relationship of the moving coordinate system relative to the reference coordinate system: the Euler angle method (also known as the three-parameter method), the quaternion method (also known as the four-parameter method), the direction cosine method (also known as the nine-parameter method), and the equivalent rotation vector method. Since the directional cosine matrix method is easiest to understand, the attitude updating algorithm of the directional cosine matrix is introduced here.

• 1. Directional cosine matrix differential equation

The relation between the relative derivative and absolute derivative of the vector is

$$ \left. {\frac{d\varvec{r}}{{dt}}} \right|_{n} = \left. {\frac{d\varvec{r}}{{dt}}} \right|_{b} + \boldsymbol{\omega}_{nb} \times \varvec{r} $$

(8.15)

When the amplitude of vector r is unchanged, $\left. {\frac{d\varvec{r}}{{dt}}} \right|_{b} = 0$, put this formula in the geographic coordinate system (n), and represented in the form of a matrix as follows:

$$ \dot{\varvec{r}}^{n} = \left[ {\boldsymbol{\omega}_{nb}^{n} \times } \right] \varvec{r}^{n} $$

(8.16)

In the formula:

$$ \left[ {\boldsymbol{\omega}_{nb}^{n} \times } \right] = \boldsymbol{\omega}_{nb}^{nK} = \left[ {\begin{array}{*{20}l} 0 & { - \omega_{nbx}^{n} } & {\omega_{nby}^{n} } \\ {\omega_{nbz}^{n} } & 0 & { - \omega_{nbx}^{n} } \\ { - \omega_{nby}^{n} } & {\omega_{nbx}^{n} } & 0 \\ \end{array} } \right] $$

(8.17)

Formula (8.17) is an antisymmetric matrix of the component of the angular velocity of rotation of the body coordinate system relative to the geographic coordinate system along the axis of the geographic coordinate system, usually expressed by symbols $\boldsymbol{\omega}_{nb}^{n} \times$ or $\boldsymbol{\omega}_{nb}^{nK}$. Each element in $\boldsymbol{\omega}_{nb}^{n} \times$ is the rotation angular velocity of the body coordinate system relative to the navigation coordinate system and can be obtained by compensating the rotation angular velocity of the reference system from the gyro measurement data.

The vector transformation formula is as follows:

$$ \varvec{r}^{n} = C_{b}^{n} \varvec{r}^{b} $$

(8.18)

Derivative on both sides of the formula, and because $\dot{\varvec{r}}^{b} = 0$, then get:

$$ \dot{\varvec{r}}^{n} = \dot{\varvec{C}}_{b}^{n} \varvec{r}^{b} + \varvec{C}_{b}^{n} \dot{\varvec{r}}^{b} = \dot{\varvec{C}}_{b}^{n} \varvec{C}_{n}^{b} \varvec{r}^{n} $$

(8.19)

Compared with Formula (8.16):

$$ \dot{\varvec{C}}_{b}^{n} = \left[ {\boldsymbol{\omega}_{nb}^{n} \times } \right] \varvec{C}_{b}^{n} $$

(8.20)

The fourth-order Runge‒Kutta method can be used to solve the above equation, and the attitude update matrix $\varvec{C}_{b}^{n}$ of the vehicle can be obtained.

• 2. Comparison of attitude update algorithms

The comparison of the above four methods is shown in Table. 8.1.

Table. 8.1 Comparison of attitude update algorithms

8.3.3 Types of SINSs

The hardware composition of SINS is basically the same, but according to the different navigation coordinate systems, the control scheme is also different. This is because according to the working principle of the accelerometer, for SINSs, the specific force measured by the accelerometer should be $\overline{\varvec{f}}_{ib}^{b}$, which is the specific force of the vehicle relative to the inertia space expressed in the body coordinate system. It needs to be converted to obtain the relative acceleration to the navigation coordinate system. According to the working principle of the gyroscope, the angular velocity measured by the gyroscope should be $\overline{\boldsymbol{\omega}}_{ib}^{b}$, which is the angular velocity of the body relative to the inertia space expressed in the body coordinate system and cannot be directly used for $\boldsymbol{\omega}_{nb}^{n} \times$ in Formula (8.21). It needs to be further calculated to obtain the angular velocity of the vehicle relative to the reference coordinate system. Based on Newton’s kinematics analysis, the following is shown:

Gyroscope measurement = angular velocity of the vehicle relative to the reference system + angular velocity of the reference system relative to the inertial space.

Similar to GINS, SINS also has analytical style, semianalytical style and geometric style. The common coordinate systems used in SINSs are the inertial coordinate system, earth coordinate system and geographic coordinate system.

Fig. 8.10

Flow chart of SINS taking the inertial coordinate system as the navigation coordinate system

8.3.3.1 SINS Taking Inertial Coordinate System as Navigation Coordinate System

The reference coordinate system of SINS is inertial system (i system) as the navigation coordinate system, then there are:

$$ \left\{ {\begin{array}{*{20}l} {\overline{\boldsymbol{\omega}}_{ib} = \overline{\boldsymbol{\omega}}_{ib} } \\ {\overline{\boldsymbol{a}}_{ib} = \overline{\boldsymbol{a}}_{i} = \overline{\boldsymbol{f}}_{ib} + \overline{\boldsymbol{G}}} \\ \end{array} } \right. $$

(8.21)

That is, the angular velocity measured by the gyroscope (the right side of equation $\overline{\boldsymbol{\omega}}_{ib}$) is the attitude change rate of the vehicle (the left side of equation $\overline{\boldsymbol{\omega}}_{ib}$). The relative acceleration of the vehicle is the specific force measured by the accelerometer compensated by the gravitation vector (the relative acceleration is the absolute acceleration under the inertial coordinate system). The velocity of the vehicle can be obtained by integrating the acceleration of the vehicle once, and the position of the vehicle can be obtained by integrating the velocity of the vehicle again. Its solution process is shown.

It is easy to solve when INS takes the inertial system as the navigation coordinate system, but its attitude, velocity and acceleration are relative to inertial space, so it is generally used in spacecraft flying away from the earth. Once applied to the near-surface vehicle, the corresponding attitude projection transformation is needed.

8.3.3.2 SINS Taking the Earth Coordinate System as the Navigation Coordinate System

The angular velocity measured by the gyroscope includes the rotation angular velocity of the earth and the rotation angular velocity of the vehicle relative to the earth. Only the latter can be used to solve the attitude matrix when the earth coordinate system is chosen as the navigation coordinate system. For accelerometers, because the reference coordinate system (earth) rotates at a constant angular velocity, the specific force is compensated by gravitation and extracted with Coriolis acceleration and translational acceleration to obtain the relative acceleration, which can be used in the velocity update and position update equations.

$$ \left\{ {\begin{array}{*{20}l} {\overline{\boldsymbol{\omega}}_{eb} = \overline{\boldsymbol{\omega}}_{ib} - \overline{\boldsymbol{\omega}}_{ie} } \\ {\overline{\varvec{a}}_{eb} = \overline{\varvec{f}}_{ib} - 2\overline{\boldsymbol{\omega}}_{ie} \times \overline{\varvec{v}}_{eb} + \overline{\varvec{g}}} \\ \end{array} } \right. $$

(8.22)

The first formula of (8.22) can be deduced directly from relative motion analysis. The second formula of (8.22) can be deduced according to Formula (8.9). Let the angular velocity of the navigation coordinate system relative to the Earth coordinate system be 0, then obtain the answer. The velocity of the vehicle can be obtained by integrating the relative acceleration of the vehicle once, and the position of the vehicle can be obtained by integrating the velocity of the vehicle again. Its solution flow is shown in Fig. 8.11.

Fig. 8.11

Flow chart of SINS taking earth coordinate system as navigation coordinate system

When using the Earth coordinate system as the navigation coordinate system, the calculation is relatively complicated because of the rotational angular velocity in the reference coordinate system. However, the attitude, velocity and acceleration calculated based on the Earth coordinate system are all relative to the Earth, which is suitable for vehicles such as Earth satellites and other Earth-orbiting motions.

8.3.3.3 SINS Taking Geographic Coordinate System as Navigation Coordinate System

When choosing the local geographic system as the navigation coordinate system, the rotation of the earth and the rotation of the geographic coordinate system relative to the earth must be taken into account at the same time.

$$ \left\{ {\begin{array}{*{20}l} {\overline{\boldsymbol{\omega}}_{nb} = \overline{\boldsymbol{\omega}}_{ib} - \left( {\overline{\boldsymbol{\omega}}_{ie} + \overline{\boldsymbol{\omega}}_{en} } \right)} \\ {\overline{\varvec{a}}_{nb} = \overline{\varvec{f}}_{ib} - \left( {2\overline{\boldsymbol{\omega}}_{ie} + \overline{\boldsymbol{\omega}}_{en} } \right) \times \overline{\varvec{v}}_{eb} + \overline{\varvec{g}}} \\ \end{array} } \right. $$

(8.23)

In the formula, $\overline{\omega }_{en}$ is the angular velocity of rotation between the navigation coordinate system and the earth coordinate system. According to the motion analysis of ships, when the navigation coordinate system adopts the ENT coordinate system, the eastward and northward velocities are V_E and V_N, respectively:

$$ \left\{ {\begin{array}{*{20}l} {\boldsymbol{\omega}_{enx}^{n} = - \frac{{\varvec{V}_{N} }}{R}} \\ {\boldsymbol{\omega}_{eny}^{n} = - \frac{{\varvec{V}_{E} }}{R} + \boldsymbol{\omega}_{e} \cos \varphi } \\ {\boldsymbol{\omega}_{enz}^{n} = - \frac{{\varvec{V}_{E} }}{R}tg\varphi + \boldsymbol{\omega}_{e} \sin \varphi } \\ \end{array} } \right. $$

(8.24)

The velocity of the vehicle can be obtained by integrating the relative acceleration of the vehicle, and the position change of the vehicle can be obtained by integrating the velocity of the vehicle again. The calculation process is shown in Fig. 8.12. The geographic coordinate system is suitable for vehicles moving on the Earth’s surface.

Fig. 8.12

Flow chart of SINS taking geographic coordinate system as navigation coordinate system

8.3.4 Initial Alignment of the SINS

8.3.4.1 Types of Initial Alignment Methods

The key problem of the initial alignment of SINS is to determine the initial transformation matrix from the body coordinate system to the geographic coordinate system $C_{b}^{n} \quad (0)$ with certain accuracy in a given time. The initial alignment of SINS can be roughly classified as follows [28]:

• (1) Classified as autonomous alignment and non-autonomous alignment based on whether the external information is used or not.

Autonomous alignment is a method to achieve alignment automatically based on the inertial sensors of the SINS itself. The method of non-autonomous alignment depends on an external reference. Autonomous alignment enhances the autonomy and concealment of SINS, but it requires far more time than non-autonomous alignment. Non-autonomous alignment can rely on external references, which can shorten the alignment time but requires support from external facilities.

• (2) Classified as coarse alignment and precision alignment based on different stages.

Coarse alignment can directly estimate the attitude transformation matrix from the body coordinate system to the geographical coordinate system by using the measurement by gravity vector g and the self-rotation rate of the earth ω_ie or by means of transfer alignment or optical alignment. The time consumption of coarse alignment is short, and the azimuth and horizontal misalignment angles can be estimated within a certain accuracy range. The dual-vector alignment method introduced in Chap. 5 is a common coarse alignment method. Precision alignment is mainly based on the gyrocompass effect, integrated navigation methods and many other ways to achieve accurate correction and calculation of small misalignment angles between reference coordinate systems and real reference coordinate systems. It provides precise initial conditions for navigation calculation by establishing an accurate initial transformation matrix $\varvec{C}_{b}^{n} (0)$. The time of precision alignment is far greater than that of coarse alignment, and the accuracy is higher. The accuracy of precision alignment has a great influence on the accuracy of INS.

• (3) Classified as static base alignment and dynamic base alignment based on the base motion state.

The attitude, position and velocity of the vehicle in the ideal state during the alignment of the static base are all 0. At this time, the interference of alignment is small, and the accuracy is high. Dynamic base alignment is more difficult than static base alignment because of the interference angular velocity and acceleration of vehicle motion. To overcome the influence of the motion environment, it is generally necessary to introduce external information and simultaneously restrict the motion of the vehicle.

8.3.4.2 Precision Alignment Based on the Gyrocompass Effect

INS usually achieves precision alignment by means of “horizontal leveling + azimuth alignment based on the gyrocompass effect”, which is called gyrocompass alignment. In the process of horizontal alignment, the horizontal misalignment angle causes velocity error by coupling with gravity, which can be controlled to achieve horizontal alignment. The process of azimuth alignment is similar to that of the gyrocompass and stabilized gyrocompass and achieves azimuth alignment based on the gyrocompass effect. Taking the eastward velocity channel of INS as an example, here briefly introduces the idea of gyrocompass alignment. To deepen the understanding of the gyrocompass alignment from the comparison of GINS and SINS, we first introduce the alignment principle of the east velocity channel of GINS. In fact, this principle is also the basic principle of a stabilized gyrocompass (see Chap. 7). Then, based on this, the alignment principle of SINS is further introduced, which is also the main working principle of the ARHS system.

• (1) Alignment of GINS

For convenience of analysis, let the vehicle be stationary and the misalignment angle be $\boldsymbol{\phi} = [\begin{array}{*{20}l} {\phi_{x} } & {\phi_{y} } & {\phi_{z} } \\ \end{array} ]^{T}$. The misalignment angle between the geographic coordinate system and the ideal geographic coordinate system tracked by the SINS “mathematical platform” is defined in Fig. 8.13. Let the gyroscope drift be $\boldsymbol{\varepsilon} = [\begin{array}{*{20}l} {\varepsilon_{x} } & {\varepsilon_{y} } & {\varepsilon_{z} } \\ \end{array} ]^{T}$, accelerometer bias be $\boldsymbol{\nabla} = [\begin{array}{*{20}l} {\nabla_{x} } & {\nabla_{y} } & {\nabla_{z} } \\ \end{array} ]^{T}$, and speed error be $\delta \varvec{V} = [\begin{array}{*{20}l} {\delta V_{E} } & {\delta V_{N} } & {\delta V_{T} } \\ \end{array} ]^{T}$.

Fig. 8.13

Definition of platform misalignment angle

Under the condition of a stationary base and small misalignment angle, the velocity error and misalignment angle equation of the eastward passage can be approximated as follows:

$$ \left\{ {\begin{array}{*{20}l} {\delta \dot{V}_{E} = - \phi_{y} g + \nabla_{x} + \dot{V}_{E} } \\ {\dot{\phi }_{y} = - \frac{{\delta V_{E} }}{R} + \varepsilon_{y} } \\ \end{array} } \right. $$

(8.25)

From (8.25), a block diagram of the eastward horizontal loop is shown in Fig. 8.14.

Fig. 8.14

Block diagram of the eastward horizontal loop

According to the principle of automatic control, the eastward channel is critically stable, and the misalignment angle does not converge. To converge the misalignment angle, the third-order horizontal leveling loop of the east channel is usually adopted for gyrocompass alignment (Fig. 8.15).

Fig. 8.15

Block diagram of the third-order horizontal leveling loop in the eastern channel

In the third-order horizontal levelling loop, by introducing node K₁ to achieve velocity feedback and damping, adding node K_E to adjust the scale coefficient proportionally, and adding the integral node K_U/S to eliminate the constant error caused by gyro drift and platform azimuth misalignment, the steady-state error of the horizontal misalignment angle ${\phi }_{y}$ will be $-{\nabla }_{x}/g$. The comparison between Figs. 8.14 and 8.15 shows that introducing three nodes, K₁, K_E, and K_U, will change the control of the angular velocity along the east direction of the platform. Through the same method, it can control the north channel and azimuth channel, obtain ω_cE and ω_cT, and achieve accurate alignment of the inertial platform.

• (2) Gyrocompass alignment method of SINS

SINS replaces the real platform with a mathematical platform calculated in the computer. Therefore, the gyrocompass alignment method of GINS can be transplanted into the SINS. That is, the signal flow of stabilized gyrocompass alignment can be used to control the platform movement, which is realized by a mathematical method. The algorithm principle of the mathematical platform is shown in Fig. 8.16.

Fig. 8.16

Algorithm principle of the SINS mathematical platform

As seen in Fig. 8.16, $\varvec{C}_{b}^{\hat{n}}$ is the computed strapdown matrix, and ${\boldsymbol{\omega}}_{ib}^{b}$ and $\varvec{f}^{b}$ are the measurements of the gyroscopes and accelerometers, respectively. ${\boldsymbol{\Omega}}_{ib}^{b}$ is the anti-symmetric array of ${\boldsymbol{\omega}}_{ib}^{b}$, and ${\boldsymbol{\Omega}}_{ie}^{\hat{n}}$ is the anti-symmetric array of the projection of the earth rotation angular rate ${\boldsymbol{\omega}}_{ie}^{\hat{n}}$ in the computational navigation coordinate system, where ${\boldsymbol{\Omega}}_{c}^{\hat{n}}$ is the anti-symmetric array of the mathematical platform modified angular rate vector ${\boldsymbol{\omega}}_{c}$ projected in the computational navigation coordinate system. $\varvec{f}^{\hat{n}}$ is the output of the transformation of $\varvec{f}^{b}$ by calculating the strapdown matrix $\varvec{C}_{b}^{\hat{n}}$.

The alignment rule of the GINS can be transplanted to the SINS if ω_cE,ω_cN and ω_cT are equivalent to the control angular velocity ω_c that is mathematically introduced. As shown in Fig. 8.17, by processing the accelerometer specific force, constructing the modified angular rate vector ω_cN, and introducing it into the attitude update calculation as a control quantity, the misalignment angle can gradually converge and finally reach stability, similar to the stabilized gyrocompass.

Fig. 8.17

Principle diagram of the horizontal alignment of the eastern channel

Figure 8.18 can also introduce a modified angular rate vector for the northward and azimuth channels ω_cE and ω_cT. If it is used as a control variable in the attitude updating calculation, the misalignment angle will converge gradually and finally reach stability, similar to GINS.

Fig. 8.18

Principle diagram of azimuth alignment

8.3.5 Basic Characteristics of SINS

For GINS and SINS, their working principles are basically the same. The biggest problem with INS is that its positioning error accumulates with time, and the long-term stability of system accuracy is poor. To improve the accuracy of INS, many countries have invested much in the development of high-precision INS. Compared with GINS, SINS often has the advantages of fast start-up, strong mobility adaptability, small size, good fitness, all-solid-state design and high reliability. Generally, the characteristics of SINS are as follows:

• (1) Because IMUs are directly attached to the vehicle, they measure the inertial linear acceleration along the axes of the body coordinate system and the angular rate of rotation around the axes of the body system, which is different from GINS. In GINS, the output of the gyroscope and accelerometer is used to stabilize the platform coordinate system in the navigation coordinate system, and the accelerometer is installed on the platform. Therefore, the accelerometer can directly measure the acceleration of the vehicle along the axis of the platform system (navigation coordinate system). In addition, because the stabilized mechanical platform is stable in the navigation coordinate system, when the navigation coordinate system is a geographic coordinate system, each axis of the platform can directly output the attitude angle of the vehicle. In SINS, the output of the accelerometer must be converted to a navigation coordinate system, and then the navigation parameters can be calculated. The output of the gyroscope is used to establish and modify the mathematical platform (navigation coordinate system) and to calculate the attitude angle. Complex electrical and mechanical platforms are completely replaced by computer software functions, which is the most important feature of SINSs.

• (2) Compared with GINS, SINS can provide more navigation and guidance information, such as attitude angular rate and acceleration measurement. The RLG SINS used in Boeing aircraft 757 and 767 can provide up to 35 kinds of information.

• (3) Although the form of the error equation is basically the same, the error propagation characteristics of SINS are different from those of GINS with different trajectories.

• (4) Because the physical platform is omitted, the size, weight and cost of the SINS are greatly reduced, and the reliability is improved. Because the IMU only plays the role of transferring information and has no feedback control for the motor, it is an open-loop control. All signal processing is realized in the computer, so it is convenient to realize. It is easy to realize miniaturization and low cost.

• (5) IMUs are easy to install, maintain and replace and to adopt redundancy configurations to improve the performance and reliability of the system.

• (6) The IMU is fixed on the vehicle, which directly bears the vibration and impact of the vehicle, and the working environment is tough. Therefore, inertial components in SINS need to have higher impact and vibration resistance. The dynamic environment of the vehicle causes great errors in inertial instruments. Inertial instruments, especially gyroscopes, directly measure the angular motion of the vehicle; for example, the maximum angular velocity of a high-performance fighter reaches 400°/s, while the lowest is likely to be as low as 0.01°/h. In this way, the range of the gyroscope is as high as that of the gyroscope with order 108.

• (7) The gyroscope of the GINS is installed on the platform and can be tested arbitrarily relative to the acceleration of gravity and the angular velocity of the Earth’s rotation, which is convenient for error calibration. The strapdown gyroscope does not meet this requirement, so it is difficult to calibrate. It can only take the IMU off the vehicle and calibrate it in the laboratory. In the course of use, depending on the stability of instrument performance or using external information, calibration can be carried out during vehicle movement.

• (8) Because SINS components are directly mounted on the vehicle and directly sensitive to complex motions such as sway and oscillation of the vehicle, the working environment is harsh, and there are many disturbances, so there will be errors such as paddling and coning under highly dynamic conditions. This kind of error will cause large errors when the vehicle is maneuvering, so the real-time compensation of dynamic errors is a problem that must be solved for high-performance SINS.

• (9) In SINS, the computational load of the computer is much larger than that of GINS, and the word length and speed requirements of the computer are much higher. Due to the amazing development of computer technology, this requirement has been solved, and 32-bit microcomputers or microprocessors can meet the requirement.

This chapter introduces two kinds of autonomous positioning methods based on the vehicle itself. Compared with direct positioning systems, it does not need to observe external references, is not restricted by weather and geographical conditions, and has strong confidentiality. It can work continuously, has a high update rate and low short-term noise. Because it does not have any optical and electrical connections with the outside world, it is especially suitable for military purposes. It overcomes the disadvantage of observation navigation based on external information. However, there are some limitations in these autonomous navigations: position parameters must be initialized, and errors accumulated resulting in position errors increasing with time. Therefore, after a period of navigation, other means are needed to help the system re-calibrated. In integrated navigation systems, direct positioning measurements are often used to correct the results of dead reckoning navigation and compensate for the error of inertial sensors.

Appendices

Appendix 8.1: Questions

• (1) Through data retrieval, please briefly describe the dead reckoning methods and characteristics used in automobile navigation.

• (2) Please briefly describe the basic method of ship dead reckoning and the main influencing factors that need to be considered.

• (3) Please briefly describe the basic principles, features and application fields of inertial navigation. Considering the characteristics of INS, please analyze the importance of inertial navigation.

• (4) What are the different technical characteristics and difficulties of long operation-time INSs and short operation-time INSs?

• (5) What are the types of gimballed inertial navigation systems, and what are their application characteristics?

• (6) What are the core components of the INS? Summarize the connections and differences of inertial sensor, inertial combinations, and inertial measurement units? Please briefly describe the types and characteristics of the gyroscopes and accelerometers.

• (7) What are the similarities and differences in the system composition, working principle and performance characteristics of gimballed inertial navigation systems and SINSs?

• (8) Please briefly describe the working principle of the initial alignment of the INS, compared with the working principle of the gyrocompass and stabilized gyrocompass to see the connection.

Appendix 8.2: The Philosophy Behind the Inertial Navigation

The dead reckoning and inertial navigation principle is the core principle of autonomous navigation. The idea is to calculate accurately the motion parameters by measuring the linear and angular motions of the given carrier in real time. For instance, the carrier position can be determined through recursive calculation or integral operation. More importantly, there are many scientific and philosophical ideas behind such a principle. For example, if the motion parameters of each time interval are calculated accurately, you can calculate the parameters for days and months without any mistakes, which echoes the idea of “a journey of thousand miles begins with a small step (千里之行, 始于足下)”. Just as the saying in Tao-Te Ching goes, “all great things have small beginnings. (天下大事, 必作于细)” It is the angular speed and acceleration measured each time that determine the overall accuracy of the system.

To put it another way, the system performance is directly determined by the accuracy of inertial components (e.g., the gyro and accelerometer), despite their small size and low power consumption. This cause-and-effect relation can also be observed between one’s daily efforts and their ultimate excellence. Even small efforts can make a big difference as long as they are ongoing; meanwhile, it is also important to keep the ongoing efforts on the right track, which makes the right goal most crucial.

In addition, a gyro with high precision and a strong fixed axis is similar to a person with focus. Focus can make people rational and calm and thus keep pursuing meaningful goals, which in turn leads to breakthroughs and innovations.

Still, the mathematical basis of the inertial navigation is the kinematics calculus formula, which reveals the coexistence of “movement” and “stillness” in the world. This dialectical unity touches on the core of ontology and epistemology. Developing such a dialectical way of thinking (e.g., absoluteness vs. relativity) can help us see the world more comprehensively, objectively, and with an open mind.