Report on of Numerical Method for Navigation Applications

In this paper, there is a proper discussion about Numerical methods that can be utilized for navigation applications. The first part will discuss the proper analysis of large methods of computation that can be used in navigation.  Through these applications, the navigation system will become extremely perfect. This paper links with different numerical methods in detail and their roles in the navigation system. The subject matter will be analyzed through differential geometry. There is one problem in navigation system that it is extremely difficult to find the shortest path curves because it will solve the problem of navigating along the arc of a great circle. This paper will highlight the navigation system in detail and discuss its important application by different numerical methods. The history of stars formation will be analyzed through numerical methods. These methods also help to investigate underwater applications in detail. The next application is related to hypertext, which means these methods can be applied easily to solve problem-related to this issue. The direct Cubic Spline method is applied in the second part of this paper, and it will help to solve an important point by computing the distance that is along the shortest path. In the numerical method that is called as Direct Spline approximation, its errors are discussed in detail.

Introduction of Numerical Method for Navigation Applications

It can be noted that we are living in a digital world. Due to this, the science of navigation system is increasing day by day. Due to this technology advancement, there is a need to make changes in navigation systems that can be applied through different numerical methods. Nowadays, navigation is extremely necessary for everyone. For proper navigation, there is a need to apply numerical methods. The invention of satellites has completely changed the attention of the navigator. He is involved in controlling the pattern and turned to fix it according to the demand. The science of navigation is moving at an extremely fast rate that is due to the invention of new and latest technologies. One of them is a radio position fixing system that was used in the past. This is also one of the best examples of a navigation system. Then after this, the invention of the radar system for the army has covered all gaps of technology in the navigation system. Now, this the new age of space traveling and satellite system. Through the help of computational position, it will become extremely easy to produce a compact computer-controlled position. This upgraded navigation system has the ability to keep the observer alert about the situation; then he can easily update travel plans.

In the past, it was extremely difficult to judge the classical navigation methods that will help to support a new technology of navigation system. There are different calculus techniques applied to find the correct navigation system for the observer. But the precise results are not obtained through such method it is only restricted to approximation. But after the use of numerical methods for the application of the navigation system is supporting the solution for this problem. Due to such action, numerical methods are used in many navigation systems. Due to this, much mathematical analysis has been reviewed and also improved for the future.

For the navigation system, there are new numerical methods has been introduced that is supporting application in the navigation system. For these applications, there are some methods of differential geometry also discussed. There are some important mathematical models that are used for navigation, and also it can be seen that this is one of the most important approaches to the study. In the past, these methods were used to make spherical model through differential geometry. For the navigation system, the spherical model of the earth has been produced, but this cannot be applied for the precise system. Through this model, future researchers are able to take the idea of making a precise navigation system.

Then after this, a correction was made in the arc of a loxodrome this can be obtained through the help of this formul

This middle latitude correction was solved through the table by a number of the different new edition. One of the most important theory is working behind this method that is called as Middle Latitude Sailing. After some time this formula is upgraded, and it will be converted into this form. For a proper navigation system, the differential geometry methods are used in detail.

Problem statement of Numerical Method for Navigation Applications

The main problem in this research is to investigate new numerical methods for applications of the navigation system. If these methods are applied in perfect order, then the navigation problem can be solved easily. The next problem is related to hyperlink for the search engine navigation system. There is a need for proper navigation system for the oceans. For this proper, there is a need to apply the numerical method for the solution.

Research questions of Numerical Method for Navigation Applications

·         How to apply navigation system through numerical methods?

·         How to apply methods of differential geometry?

·         How to find a precise history of the star’s movement?

·         How to analyze navigation in hypertext?

Research aims and objectives of Numerical Method for Navigation Applications

The main aim of this research is to highlight different navigation applications that can be applied through numerical methods. The next purpose of this study is to study the data to determine the star history in a quite precise manner through navigation. The last aim of this research is to analyze user movement in hypertext by numerical method

Literature review of Numerical Method for Navigation Applications

According to the author Williams (2000), it is conducted that by a great deal of the energy as well as enthusiasm, a science of the navigation is entered in the electron age. For the orbiting, the man-made satellites have the large extent which is replaced in a stars attentions for the modern navigation along with the orbits of the various satellites as compared to the rotation of the heavens that tends to control the pattern for the positions of the observer which is fixed in the day. In this research, the main focus is to take the opportunity for the introduction of the numerical methods for our own that are found particularly as well as useful application in the navigation. The numerical methods in the navigation application are paying for the attentions underlying where the mathematical model is used. There is no written exposition in the navigation’s that also adopts the rigorous approach of mathematical to the subject. In this research adopted the two models for a shape of the earth, the first one is the spherical model as well as the 2^nd one the spheroidal model which present the navigation. Whereas the spherical model is not the bad approximation that is applied for the various problem in a navigation’s which is used consistently .For the observer traveling the correction which is applied to the Mean latitude and it also gives the Middle latitude with the arc of the palindromic as it computed the middle latitude which is determined the by using the below formula;                     

Whereas the difference of the latitude is the number of the minutes per arc on the sphere of the meridian of the sphere, as compared to the difference of the meridional part used the complied as mention in the above formula (Williams, 2000).

Need for a Numerical Method

According to the author Weintrit (2011), it is conducted that the need of the numerical methods in the navigation application is very important. In the below figure a general polygonal boundary in a Z-plane is shown, which is transformed from an infinite straight line where the t-plane like the upper half of the t-plan transform interiors of the polygonal through this equation;

Navigation systems of Numerical Method for Applications

According to the author Syed & et al. (2007), it is conducted that, the worldwide navigation systems if GPS (Global positioning systems) and these systems always need the clear line of the sights for the orbiting to satellites. The clear line of the sight is not determined, for the vehicle navigation’s at all the time of the vehicle could travel by tunnel below the bridges , and the forest of canopies in the urban canyons. Consequently, the numerical methods are the perfect candidate for the integration of the navigation applications by GPS. Whereas the sensors need to use the accurately as well as reliably bridge of the GPS signal gaps, where the multi-position for the calibration is designed for the numerical methods or medium quality. The definition of the navigation refers to the best estimation of the moving objects in the different terms of the velocity, positions as well as attitude of the moving objects. When the globally available the GPS provide the high navigation’s of the accuracies at the very low cost , where the GPS also receives the very portable  and the lower consumption if the integration of the sensor (Syed & et al., 2007).

Nonlinear Filtering with Applications to Navigation

It is reviewed by the author Sadist (2001), the nonlinear filtering technique is also included in the numerical methods. So the numerical method in the filtering techniques is the estimation along with the detection of that arises which is in a satellite that based on navigation. The numerical experimentation in the projectile particle filtering is exceeded the filtering methods of the navigation performances. Now for the positioning system, the reliability is very significant for the navigations process. Thus the monitoring system of the integrating is the inseparable part for the navigation systems. Due to the malfunctions, the failure changes in the sensors as well as actuators that must be detected along with to repaired the integrity of the systems. Several methods of the navigation’s that is also used for the carrier phases of the GPS differential, and it must be able to isolate as well as to detect the ship's systems whenever they occurred. In the signal loss, the time period is very long, so the position of the information is required by the GPS receivers where the signal has the three fewer of the satellite, and the numerical techniques are solely based on the navigation’s applications. In this research study, the numerical method for the approximation techniques is totally based on the navigation systems. In the signal, the loss is sufficiently for the long-time period, and the position is needed for this of the GPS receivers. The numerical method allows using the carrier as part of the navigation in the formations (Sadjadi, 2001).

According to the author Jun & et al. (2014), it is conducted that for the insufficient application for the navigation systems by the use of the numerical method is new for the navigation techniques and it is composed through the SINS as well as the infrared magnetometer system. When the user of the integrated navigation system is proposed, then it could also improve the navigation which is easily achieved. Currently, the application of the navigation system for the numerical method is the SINS for the navigation systems, and the advantages are that it gives the accuracy of the navigation systems and it also has the strong anti-interference ability. Thus the use of the navigation satellite is shortcomings in an actual process, which is also evident for the lack of autonomy along with the over-reliance on the satellite. In the numerical methods, which are the indirect methods, the error of a navigation parameter is then integrated by the navigation systems is the main statuses for the filter valuations in the parameters of the navigation. The states of the error are inertial of the navigation systems are estimated through the Numerical methods along with the calibrated results. The positioning accuracy could be improved largely by the Numerical methods compared with the SINS only. Due to the malfunctions, the failure changes in the sensors as well as actuators that must be detected along with to repaired the integrity of the systems (Jun & et al., 2014).

According to the author Dimov & et al. (2011), it is conducted that, by using the regular numerical methods which is mention as in the other studies it solves the convection of the dominated problem of the navigation systems. Whereas the numerical error always occurs and the oscillations around the computed profile are the smearing of the concentration units. Then it also decreases the numerical error. The attention of the numerical methods by the respect of the previously mentioned for the different limitation of the navigation systems. The parameters which are distributed for the model of mass in the navigation and the pours of media is given in the other views. The numerical methods developed to deal with the advection of the dominated navigations issues whereas the upstream of the weighting FE, and the method of characteristics for the navigation if modified. The model of the numerical method for the navigation application is generally intractable analytically due to the irregular geometry as well as the non-homogeneous structure if the flow of region. The numerical method is required for the practical application of the numerical solutions, and it is limited for the 1-D and the 2-D cases to avoid the high expense of the 3-D model in the human as well as the machine resources (Dimov & et al, 2011).

Method of Numerical for Navigation Applications

            For the study of navigation, the simplest way to find the shape of the earth is to consider the sphere. For the researcher, it is a reasonable approximation, but it contains some huge errors. This is because it can be seen that the real shape of the earth is not spherical. Moreover, some problems occur when the shortest path was required. For that case, there is a need for simplest method, and through this, it will become easier to find its shape, and that is an oblate spheroid. It contains a small axis, and that coincides with the axis of revolution of the earth. It can be seen that the axis of the earth mainly considered as sphere or spheroid. But many research considered spheroid for the exact shape of the earth. There are some flat surfaces on the earth, and these can be found easily through this formula

In this equation, a is known as the semi-major axis of the ellipse and b is known as the minor axis. For that case, different locations of the earth are defined by one spheroid, so it is better to say that the Earth is a smooth union of different spheroids. By considering these facts, the navigation system can be easily modified with perfection. One of the best numerical methods is used for defining a position on the surface of the earth. This can be done by the intersection of two big circles that are called Great Circle, and other circles that are not the part of the Great circle is known as Small Circle. For the proper application of the navigation system, many researchers are using the example of the shape of the earth. All distances on the earth surface can be expressed in the long-form that is for defining the arc of the equator. This distance is also known as Geographical Mile, and it is used for the treatment of navigational methods.

The method that is used for the Velodrome curve that is present on the earth is important to be considered as a navigation system. That is basically like a THUMB line that is present on the sphere. Through the use of the numerical method, it will become easy to find the length of velodrome form any point on the equator. This can be found through this formula

In this equation, a is the radius.

The next application of the navigation system is related to underwater. This can be done through the way-point calculation in numerical methods. But from them, one of the best methods is way-point because it will help to form a reference trajectory. Through the help of the controller, the control signals are generated. The sensors that are used to navigate the underwater surface can be done through numerical methods. For monitoring a map, aided navigation method is used to monitor the performance, and that method is completely based on the radar system. For surface-navigation, a radar system is used. These sensors are involved in measuring geometrical distance through numerical methods. Thus, the measuring relation can be found through this formula

Limitations of Numerical Method for Navigation Applications

·         The limitation of this method is that it grows the complexity for the high dimensional systems, and for the small error.

·         The numerical method is not always implemented in the real-time application of the navigation

·         The numerical method is not well suited for the realities of the inertial navigation systems for the data streams by the data gaps.

·         The algorithm of the numerical method is very complicated for the navigation system where the data is very limited for this algorithm.

·         Variable step-size predictor-corrector for the numerical methods of the navigation system is used it overcome these limitations.

·         The numerical method has its limitations; like the computations of the multi path where the error is an envelope for the assessment of the navigation signals, and the approximations of the numerical method is the multi path error.

·         To overcome these limitations get the more realistic view as well as the actual multi path for the different performance where the simple model is supposed to be statistical for the distribution of the geometric  paths, and the delays of the multi path are the relative for the geometric distributions

Conclusion  on Numerical Method for Navigation Applications

Summing up all the discussion from above, it is concluded that numerical methods play a very important role in the navigation system applications. Through these methods, it will become easy for the modified navigation system. In this paper, there is a deep discussion of some numerical methods used for the navigation system. The subject matter will be analyzed through differential geometry. There is one problem in navigation system that it is extremely difficult to find the shortest path curves because it will solve the problem of navigating along the arc of the great circle due to this technology advancement there is need to make changes in navigation systems that can be applied through different numerical methods.In the past, it was extremely difficult to judge the classical navigation methods that will help to support a new technology of navigation system. There are different calculus techniques applied to find the correct navigation system for the observer.

 For these applications, there are some methods of differential geometry also discussed. There are some important mathematical models used for navigation, and also it can be seen that this is one of the most important approaches to the study that case there is a need of more simplest method and through this, it will become easier to find its shape, and that is an oblate spheroid. It contains a small axis, and that coincides with the axis of revolution of the earth. For that case, different locations of the earth are defined by one spheroid, so it is better to say that the Earth is a smooth union of different spheroids. The method that is used for the Velodrome curve that is present on the earth is important to be considered as a navigation system. The next application of the navigation system is related to underwater. This can be done through the way-point calculation in numerical methods.

References of Numerical Method for Navigation Applications

Dimov, I., & et al. (2011). Numerical Methods and Applications. Lecture Notes in Computer Science. doi:10.1007/978-3-642-18466-6

DolphinP, A. (2002). Numerical methods of star formation history measurement and applications to seven dwarf spheroidals. Monthly Notices of the Royal Astronomical Society, 332(1), 91–108. doi:10.1046/j.1365-8711.2002.05271.x

Jun, W. J., & et al. (2014). Analysis of application of a new integrated navigation system of the tactical missiles. Proceedings of the 33rd Chinese Control Conference, 653-657. doi:10.1109/chicc.2014.6896702

Sadjadi, B. A. (2001). Approximate Nonlinear Filtering with Applications. Professor P. S. Krishnaprasad: Department of Electrical and Computer Engineering. Retrieved from http://sci-hub.tw/https://apps.dtic.mil/docs/citations/ADA439809

Syed, Z., & et al. (2007). A new multi-position calibration method for MEMS inertial navigation systems. Measurement Science and Technology,, 18(7), 1897–1907. doi:10.1088/0957-0233/18/7/016

Weintrit, A. (2011). Methods and Algorithms in Navigation: Marine Navigation and Safety of Sea Transportation. CRC Press.

Williams, R. (2000). Applications of numerical analysis in navigation. PhD thesis The Open University.