1. Abstract Currently, few navigation systems used by aircraft are truly adaptable to all-weather, global applications. Traditional radio navigation, such as TACAN, has many limitations and inconveniences in its application. To overcome this shortcoming, an inertial navigation system (INS), which provides all navigation data for pilots without the need for external auxiliary devices, has been successfully developed and widely used. However, its minute position errors accumulate proportionally to the square of the flight time, thus severely affecting navigation accuracy over long flights. Without proper correction, position errors can accumulate to over 300 meters per hour. Another sophisticated navigation system, GPS, while its errors do not change over time, is not a panacea. It has advantages but also inherent limitations. It is prone to lock-on failure when measuring highly maneuvering targets and is susceptible to external environmental and electromagnetic interference. Furthermore, GPS's short-term relative error is greater than that of INS; relying solely on it for navigation or control can have the opposite effect. Therefore, in navigation system design, inertial systems are often used in conjunction with them. GPS and INS happen to complement each other, and through a set of algorithms, the advantages of both can be retained while their disadvantages can be eliminated. This paper discusses the characteristics of these two navigation systems and uses the Kalman filter rule to derive simple measurement data relationships, designing a "GPS/INS combined navigation system." 2. Introduction Early ship navigation often relied on "navigation methods" to determine the position and direction of the vessel, observing land features to guide the ship towards a target. With the advent of aircraft, early flight also relied entirely on the pilot's sense of direction, distance, altitude, and speed to control the flight, performing takeoff, landing, and transit maneuvers. This art of controlling the direction and distance indication of a vessel from one place to another is called "navigation." However, relying solely on manual navigation methods, while visual judgment is not difficult in good weather or when there are many obvious reference points, it becomes crucial to rely on pilot experience, skill, and luck to determine direction and location when weather conditions are poor, visibility is low, or reference points are absent or unclear. This adds significant stress for pilots and introduces numerous uncertainties that could seriously impact flight safety. Therefore, various navigation technologies have been actively developed. With rapid technological advancements, the art of navigation has become more diverse, precise, and reliable. "Navigation science" can be defined as "an application of calculating and determining the position of a vehicle relative to a predetermined destination." More advanced radio navigation systems, such as Loran, VOR (Very High Frequency Omnidirectional Range), Distance Measuring Equipment (DME), TACAN, and Doppler, have been developed, effectively assisting navigators and providing vital navigational references. However, radio systems still have limitations and inconveniences; factors such as distance and terrain obstruction can cause malfunctions. Furthermore, the basic architecture of radio navigation requires a "base station" to transmit positioning radio signals, which are then received, processed, and calculated by the aircraft's "receiver" antenna to display the relationship between two points and obtain navigation data. If either the base station fails or the radio transmission is faulty, navigation cannot function, which is extremely dangerous when flying in the open air. Therefore, in the 1950s, the U.S. Department of Defense recognized the need to develop a navigation system that could provide all navigation data for pilots without the need for external auxiliary devices. At that time, MIT developed the first inertial navigation system (INS) for aircraft. This system is completely self-contained, an independent source, and unaffected by the external environment, measuring and providing all navigation data, including the vehicle's precise position, ground velocity, attitude, and heading, to the autopilot and flight instruments (such as the horizon and compass). Because the functionality, size, and weight of INS are far superior to other navigation systems, it has maintained its leading position in the navigation field in recent years. However, the position information provided by inertial navigation systems still contains a small amount of error. Although this error changes slowly, the accumulation of position error is proportional to the square of the flight time. Therefore, it affects the navigation accuracy during long-duration flights. Without proper correction, the position error can accumulate to more than 300 meters within an hour. Thus, although INS is an independent operating system, it still has shortcomings. The causes of error are related to factors such as the quality of the accelerometer and gyroscope, changes in the gravitational field, initial position, azimuth input values, and installation errors. Of course, the quality of the system itself varies greatly depending on its price. Since the main error sources of INS are the angular rate drift of the gyroscope and the deviation of the accelerometer, and these errors increase over time, if some equipment can be used to appropriately correct the errors caused by INS within a certain period, the system's navigation accuracy can be significantly improved. In the 1960s, the US Navy developed the TRANSIT navigation satellite system for ship and submarine positioning. To this day, many ground-based vehicles still use this less precise navigation satellite system. In the 1970s, the U.S. Air Force began researching and developing a three-dimensional NAVSTAR (Navigation Satellite Timing and Ranging) precision satellite navigation system. In 1973, the U.S. Defense Navigation Satellite System (DNSS), in conjunction with the Navy's New Experimental System (TIMATION) and the Air Force's "Program 621B" program, expanded this into the faster and more accurate GPS (Global Positioning System). Generally, this global positioning system calculates the receiver's position by observing GPS satellite broadcast signals. It uses two positioning methods: pseudo-range observation and carrier phase observation. While carrier phase observation is more accurate than pseudo-range observation, it faces challenges such as cycle dropout and undetermined phase values, posing significant technical obstacles and lower reliability for navigation applications. Therefore, carrier phase observation is currently mainly used for long-term fixed-point observations, such as geodesy and geodynamics; while pseudo-range observation is widely used for real-time positioning navigation. During GPS positioning, its broadcast signal is affected by many factors, such as atmospheric refraction, satellite orbital position deviations, and clock errors, which can impact its positioning accuracy. Currently, the P-code (PPS: Precise Positioning Service) used in GPS broadcasts is strictly limited due to its high accuracy; only the US military and specially authorized personnel can use it. The C/A code (SPS: Standard Positioning Service) is unrestricted, but its accuracy is lower. If Selective Availability (SA) is also enabled, the error will be further aggravated. Therefore, for real-time positioning requiring high accuracy, a system using C/A codes that can significantly improve accuracy is needed. DGPS (Differential GPS) is a system developed to improve the accuracy of GPS positioning using coded signals. It works by using the principle of relative positioning. First, a fixed GPS reference station is set up with its geographical location precisely calibrated. Then, the position is compared with the position set by the GPS receiver to find the GPS positioning error of the reference station. This error is then broadcast to the user. In this way, the accuracy of DGPS can be improved by more than ten times, reaching the meter level. However, the position accuracy of GPS at any given moment in a short period of time is still much worse than that of INS. As can be seen from the above, although GPS has the advantage of its error variation not changing with time, GPS is not suitable for high maneuverability, is prone to lock-up, and is susceptible to external environmental and electromagnetic interference. INS, on the other hand, can measure the position, velocity, acceleration, and attitude of highly maneuverable targets and is not affected by external interference. In the short term, the relative error of INS is also much smaller than that of GPS. Therefore, INS can be used to verify and modify GPS measurement results. Thus, a combined GPS/INS navigation system is a better choice, as it can obtain high-precision and high-reliability navigation information. Furthermore, in terms of filter selection, the combined GPS/INS navigation system basically adopts the Kalman filter rule because it is simple and reliable and has been widely used in GPS/INS navigation systems. [b]3 INS/GPS Basic Principles[/b] 3.1 Inertial Navigation System (INS) Basic Principles 3.1.1 INS Principles INS generally has two structures: ring-type and strapdown type. In the ring-type system, the accelerometer and gyroscope are placed on a reference platform, allowing the rotation between the sensor and the carrier to be independent, thus maintaining measurement and navigation operations in a stable coordinate system. Possible navigation coordinate systems include Earth-Centered Inertial (ECI), Earth-Centered Fixed Coordinate System (ECEF), North-East-Down (ED) coordinate system, and coordinate systems incorporating Wander angles. Ring-type systems are relatively accurate and easy to calibrate (no coordinate transformation is required; automatic northward correction is achieved using the Earth's gravitational field), but they are larger, heavier, more expensive, and less reliable. Strapdown systems, on the other hand, have sensors fixed to the carrier aircraft. Coordinate transformation is used to measure the acceleration and velocity of the moving object so that navigation calculations can be performed within the inertial system. This approach is suitable for highly maneuverable applications, and especially with the advent of new, high-quality gyroscopes and accelerometers, strapdown inertial systems will become the primary device due to improved cost and reliability. The definition of a strapdown structure is as follows: The biggest difference between strapdown and traditional ring-type systems is that inertial navigation equipment such as gyroscopes and accelerometers are directly mounted on the carrier, rather than on a reference platform. Furthermore, the onboard navigation computer can continuously track the aircraft's attitude relative to a predetermined reference inertial axis based on gyroscope signals. As a result, because the computer can provide the necessary coordinate transformations to align the accelerometer outputs with the computer's calculated reference axis—in other words, the transformation is performed analytically within the computer—the inertial reference platform in traditional systems can be replaced by the following two functions: 1) establishing the attitude inertial axis based on the gyroscope output reference; 2) converting the accelerometer outputs into inertial coordinate variables via coordinate transformation; 3) since the strapdown structure can directly provide aircraft-related signals, some devices commonly used in traditional systems can be omitted. Within an INS system, there are generally numerous error sources for evaluating system accuracy and characteristics, such as gyroscope and accelerometer-related errors, which basically include static g-sensitivity deviation and drift, size factor errors, misalignment errors, and random errors. Additional errors arise from navigation calculation corrections, initial and alignment transformations, and inaccuracy calculations. Without compensation, all INS errors change over time, while some errors (such as position) diverge with time, and others are constrained and oscillate. Therefore, the accuracy of INS is highly dependent on sensor quality, navigation system mechanisms, and aircraft dynamics. INS is generally designed for independent operation. Regarding error characteristics, since most require high accuracy, external auxiliary devices can be used to reduce INS errors. An INS with auxiliary devices uses data from various auxiliary devices (such as tracking radar, GPS, TECOM, etc.) and a navigation Kalman filter to improve the accuracy of navigation data. 3.2 Basic Principles of the Global Positioning System (GPS) Satellites orbit the Earth, regardless of whether their orbits are elliptical, circular, or geostationary, always traveling at a certain period. If there are no interfering factors (e.g., the gravitational pull of the moon and sun, uneven Earth gravity, air molecular drag, etc.), the satellite's orbit remains fixed, meaning it maintains a certain relationship with the Earth. Therefore, we can accurately calculate when, where, and during what time periods it passes through which areas. Since its operation is highly precise, people on Earth can use it for navigation. It transmits its position data relative to Earth's coordinates via radio. The spacecraft receiver, Earth, and satellite form a closed triangle. The distance between the satellite and the Earth's center is known. If we can measure the straight-line distance between the spacecraft and the satellite, we can deduce the spacecraft's coordinates relative to Earth, obtaining positioning and navigation data. This is the basic concept of "satellite navigation." In fact, the "satellite navigation" method originated from ancient people's observation of celestial phenomena to determine location, evolving naturally. Satellite navigation system structure: Taking the GPS system as an example, the entire system structure is divided into three main parts. 1) Space Segment: This mainly consists of the satellites themselves and the satellite constellation. There are a total of 24 satellites in space (21 operational, 3 backups), distributed in 6 orbits 20,200 kilometers above the Earth. Each orbit is inclined at a 55-degree angle to the others, with four satellites in each orbit. The period is approximately 12 hours. With this arrangement, four satellites can be seen at any time and place on Earth for three-dimensional spatial positioning. 2) Ground Control Segment: As the name suggests, these ground tracking stations control the normal operation of the satellites. If necessary, they can alter satellite data to cause adverse consequences for any location intending to use it for illegal activities. The control segment consists of five monitor stations and three radar communication stations distributed across free areas globally. The monitored satellite data is immediately transmitted to the Springs Master Control Station in Colorado, USA. High-speed computers calculate the orbital parameters and correction commands for each satellite, and this result is transmitted to the satellites in orbit via radar. This ensures the satellites maintain accurate status, serving as the basis for navigation. 3) User Segment: The user's equipment is simple, including a frequency band antenna, data processing unit, display components, and button unit. As long as the antenna is not interfered with or blocked, and can receive signals from three or more satellites simultaneously, it can display latitude and longitude coordinates. A satellite orbiting the Earth involves the interaction of gravity and centrifugal force between two objects, falling under Newtonian mechanics and discussed in inertial coordinates. However, since the Earth rotates on its axis and revolves around the sun, both are in motion, determining the satellite's position in Earth's inertial coordinate system becomes very complex. GPS's basic working principle is triangulation. Once the satellite's position is known through precise satellite data, the user receives the data signal transmitted from the satellite and calculates the transmission time. Because the signal travels at the speed of light, the user can calculate the distance to the satellite. This actual measurement (generally referred to as the virtual distance) usually includes errors caused by the deviation of the user's clock relative to the GPS reference clock. Because atomic clocks are used on the satellites, the error is relatively smaller than that of the user's clock. Therefore, when determining a three-dimensional spatial position, the clock deviation must be considered. At least four satellites are required to find a suitable solution for navigation. Velocity can be calculated using different methods; the final measurement result is called the distance difference. Inertial navigation determines the position and velocity of a carrier relative to Earth's inertial coordinates. The navigation solution is obtained by deriving Newton's equations of motion from the carrier's specific gravity and the rotational rate of instruments on the carrier. GPS satellite system modes contain satellite experience information related to time functions and satellite broadcast data characteristics. Given the positions of GPS satellites and their associated distances, the receiver can determine its own position, velocity, and clock deviation. Measurement data formats include virtual distance, range difference, phase data, multi-antenna phase difference data, and differential GPS data. To calculate satellite position and velocity, the receiver requires partial atmospheric data and then uses orbital interference modes to calculate the true position and velocity of the GPS satellites. The orbit of each satellite is randomly corrected for each lateral, longitudinal, and orthogonal error value. For each error value, the error magnitude and interference period are randomly selected. Corrections for major ground station location estimations and predictions are included in the broadcast data. In this estimation, any error included in this mode can be used as a bias in the calculated measurement. The most common method for measuring distance is to multiply the radio wave transmission time by the speed of light. However, using time to determine the starting point is susceptible to external interference and has poor accuracy. Another method is to calculate the number of wavelengths from the satellite to the antenna and calculate its accuracy from the phase relationship. Because the wavelength is constant, its accuracy can reach the centimeter level. This method is called "Carrier Phase Measurement". However, when the receiver tracks GPS satellites, it may sometimes lose lock due to the following reasons: signal obstruction by the host or external objects, signal weakening due to terrain or environment, interference from external transmission components, or flight attitude. The quality of GPS tracking is closely related to manufacturing, modularity, and specific receiver antenna characteristics. Therefore, as a basic assumption, four or more satellite tracking channels, Y-code virtual distance and distance difference data, and the ability to quickly and repeatedly measure distances during dynamic measurements are essential. The GPS receiver mode can calculate the tracking function between the master receiver and the tracked satellite, such as distance, rate of change of distance, and acceleration. All of the above values ​​can be estimated by calculating the difference at a frequency of once every 0.1 seconds. Whenever acceleration exceeds the tolerance threshold, the tracking satellite will lose lock. When satellite tracking fails, the original required time can be used to determine when the data will become valid again. In highly maneuverable applications, the receiver requires auxiliary data from the INS to help maintain satellite signal tracking lock. Navigation is closely related to the accuracy of the receiver's state mode and whether the GPS data strongly correlates with the terrain at each measurement correction. If the data is taken from four satellites and the dynamics are within a reasonable range, accurate position and velocity can generally be found. When using different GPS antennas, satellites on any horizontal plane can be tracked using some fairly simple gain methods. Other methods use more detailed digital models that can calculate the dynamic values ​​between the host vehicle (HV) and each tracked satellite (SV), such as virtual distance, distance difference, and acceleration. The antenna gain can be calculated from the antenna's angle of incidence to determine whether a satellite is being tracked. As for which satellite to use for tracking, it is basically based on the one with the strongest terrain structure that can produce the most accurate solution. Satellite selection involves scanning first to determine the optimal terrain for the tracking scenario. Of course, the satellite signal must be accessible to the receiver antenna, unobstructed, and have adequate signal strength on the Earth's horizontal plane. To avoid interference, satellites in certain directions must be avoided. In some applications, tracking the same satellite during flight is essential. At other times, maneuvering a combination of four satellites helps reduce the likelihood of loss of lock-on during flight. The receiver mode includes an option to simulate interference effects during GPS signal reception. Any type of interference defined by location, form, and effective radiated power can be established and input into the mode, determining the angle of incidence of the interference signal at the GPS antenna gain. Assuming the interference source is a dominant continuous wave at GPS frequencies, with fixed GPS signal strengths at L1 (-163dBW) and L2 (-166dBW), the overall effect of the interference source can be analyzed if the interference source signal can be assumed. This mode can calculate the power ratio (J/s) of each transmitter/receiver to the interference signal. The J/S ratio is a function of the radiated interference power, the distance between the receiver and transmitter, the GPS receiving frequency, and the receiving antenna gain. Interference effects on the GPS receiver should be isolated to improve tracking performance and navigation accuracy. In current models, signal reception can be handled in a simplified manner: when the J/S value exceeds certain tolerance values, such as a requirement of 45 dB and a tracking value of 65 dB, it is unavoidable that the GPS receiver must possess the ability to define requirements and track satellites. GPS is a new satellite navigation device; strictly speaking, it falls under the category of radio navigation, using radio ranging and receiving related data to determine the aircraft's position and velocity for navigation purposes. With the maturity of GPS technology and improved reliability, GPS characteristics far surpass existing radio navigation systems such as TACAN, and only inertial navigation systems (INS) can rival it. GPS navigation systems can provide authorized users with accurate three-dimensional spatial position, velocity, and time. GPS systems have a wide range of applications, are unaffected by weather, and the receivers are inexpensive, small, lightweight, require low power, and are highly reliable. Therefore, GPS is worthy of widespread application in navigation systems. 3.3 GPS/INS Integration Principle Both GPS and INS provide effective accuracy when used individually, which is undeniable. However, due to their different design logics, they have different limitations in use. GPS has shortcomings that INS lacks, and GPS can compensate for the deficiencies of INS. Overall, only by combining GPS and INS can a truly perfect navigation system be achieved. In individual applications, INS can provide continuous and accurate auxiliary data under short-term, high-mobility conditions, while GPS provides discrete and accurate auxiliary data over long periods. In other words, INS has a smaller error than GPS in short-term and immediate situations, but for long-term use, it is necessary to use discrete GPS measurements for correction. By understanding the system drift, the state parameters can be quickly estimated and converged. 3.3.1 Integration Structure Introduction The integrated system has a Kalman filter that processes the virtual distance and distance difference measurements required for satellite derivation and uses them to estimate and calculate the error margin between GPS and INS measurements. The final result is fed back to correct the INS, providing accurate navigation values. 3.3.2 Kalman Filter The Kalman filter is a back-calculation filtering method developed using state-space technology. Its characteristic is that it does not require storing past measurement data. When new data is measured, based on the new data and the estimated state parameters from the previous moment, the new estimated state parameters can be calculated using the system's own state transition equations (i.e., dynamic equations) and a set of back-calculation formulas. 4 Conclusion The basic task of an inertial navigation system (INS) is to provide navigation data that establishes a fixed coordinate system (Earth) relative to the aircraft's geometric relationship. This data serves as a precise and reliable guidance reference, and is used by the autopilot or pilot to perform track corrections. Especially for military aircraft operating deep in enemy territory, where radio silence is maintained and friendly navigation base stations cannot provide information, INS is the only reliable and unaffected system. Combined with corrections provided by GPS measurements over a certain period, and the processing of environmental clutter interference using the Kalman filter principle, a certain level of accuracy in navigation data can be maintained. Overall, combining GPS and INS aims to utilize their complementary effects, retaining their advantages and eliminating their disadvantages to obtain a perfect navigation system. Of course, these foundations are merely conclusions of scientific argumentation and logic. The final results must be put into practice through hardware equipment. This involves various human factors such as manufacturing, maintenance, and upkeep. The realization of all mathematical rules must return to people's own persistence and combination. Therefore, with the support of rigorous and correct factory and ground testing, maintenance, and adjustment, the correct navigation data can be provided after the navigation system is installed, for pilots to refer to in real time, and transmitted to other airborne electronic equipment for navigation correction calculations. The accuracy of a system is the cumulative result of each system and component. Only by controlling and implementing each link can the system function achieve the design goals, exert tactical effectiveness, and carry out precise combat strikes. [b]5 References[/b] I. Huang Guoxing (1996): Principles and Applications of Inertial Navigation Systems, Quanhua Science and Technology Books. II. Song Zhentan, Song Zhenyao, Jiang Zhonghong, Yuan Minshi (1998): Aircraft Communication and Navigation Systems, Gaoli Books. III. Liu Shaoqing and Cai Youlong (2001): Research on the Theory of GPS/INS Navigation Integration System, *New New Quarterly*, Vol. 29, No. 1. IV. Gelb Arther Ed, "Applied Optimal Estimation", MIT Press, Cambridge, Mass, 1974. V. Brison, AE Jr. and Ho, YC, *Applied Optimal Control - Optimization, Estimation, and Control*, Hemisphere Publishing Corporation, 1975. VI. Stengel, RF, *Stochastic Optimal Control - Theory and Application*, Wiley-Interscience, 1986. Editor: He Shiping