Application of the Least Squares Method in Navigational Sailing

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Project title

Code/計畫編號

MOST103-2410-H019-024

Translated Name/計畫中文名

最小平方法應用於航海計算之研究

Project Coordinator/計畫主持人

Wei-Kuo Tseng

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Merchant Marine

Website

https://www.grb.gov.tw/search/planDetail?id=8321106

Year

2014

Start date/計畫起

01-08-2014

Expected Completion/計畫迄

31-07-2015

Bugetid/研究經費

366千元

ResearchField/研究領域

管理科學
交通運輸

"最小平方法應用於航海計算之研究航海及航空領域中航行的距離長，所以要考慮到地球表面的曲率，不能再用單純的平面座標來表示距離及方位。我們在解決航行及測量問題時，通常都把地球模型般分為第一近似地球參考模型的正球體(Sphere)及第二近似地球參考模型的兩極稍扁的橢球體來協助我們解決問題。第一近似參考模型常應用在短距離範圍內使用來解決距離、位置及方位問題，在較長距離的全球性航空及航海航行問題則是以當作第二近似參考模型來解決問題的，這也是大部分 GPS全球定位系統所使用的模型。早期的導航軟體中，因為機器硬體設備的限制及其他的理由所以軟體計算時都是用有限精確度的計算方法來給航行者，以及因為計算的基準是在一個橢球或球體上，所以一定會有一些小許的誤差量，但隨著時代的進步與變遷，電子海圖顯示系統(ECDIS, Electronic Chart Display And Information System) 已經在要求精確度與連續性的定位，所以有限精確度的計算方法可能已經不符合要求了，傳統查表方式的航海已經不能很精確地配合現有電子輔助儀器的精確度，而且很多電子航海儀器只是沿用傳統的計算方法或利用包含高次方項次的展開式來提升精確度，這樣一來電腦必須要使用非常多的記憶體以及其他電子資源，隨這電子儀器的介面越來越人性化，所需要的電子資源也就越來越多，尤其以 3D立體影像的介面所需要資源最多，因此節省電子資源是一項非常重要的課題，因此有必要提出可減少電子資源消耗又符合合理之精確度的新穎的航海演算法，所以本研究將提出運用最小平方法所找出簡化公式及其係數來達到此目的。首先以最小平方法來找出地球WGS 84橢球體子午線的新長度公式，要找出子午線的新長度公式的原因是因為很多航海航法都涉及到地球的子午線長度，例如恆向線航法及大橢圓航法。我們要從WGS 84 所定義的橢球體上出發來研究地球上任一點的表示方法，並研究橢球體的幾何性質，接著應用最小平方法求取子午線長度簡化公式。子午線弧長公式可以運用數學微積分的基本定義導出線段長度的積分關係式，而因為這個積分式非常複雜幾乎不可能用人工去完成，幸好現在科技的進步有解題專用的數學軟體(例如 Mathematica, Maple及 Matlab等數學軟體)，用軟體的代數符號運算可以為我們解決這個研究所帶來的複雜積分式子，把長度積分式用成泰勒展開式(Taylor series)展開，最後依照需求展開到任意精確度，再把不同緯度的子午線長度給整理出來後直接使用，當長度積分式展開的項次很多時其精度必然相當準確，就可以此為基準以最小平方法來找出新的子午線長度簡化公式的新係數，得到一套新的子午線長度公式，日後使用者就不需要用困難的積分式與泰勒展開式等等方法來求子午線的長度，降低電子資源的需求也減少了運算的時間，因為子午線長度涉及到其他航海數學問題，因此也間接減少其他航海數學問題的電子資源與運算時間，這個方法可以推廣到子午線、大橢圓、恆向線正反解的問題上，也可以持續應用在其他的航海問題計算領域以減少所需要的電子資源並維持一定水準的精確度。本研究成果預計將建立一系列簡單、易懂又具邏輯性的計算方法，對於電子海圖顯示系統或地理資訊系統軟體設計者編寫程式語言及航行人員使用商業數學軟體計算航行問題時會有很大的助益，本研究也將撰寫相關的 Javascript程式將成果顯示在 Google Maps上，另外也可以開發相關航海問題的Matlab程式及 EXCEL增益集(因為微軟試算表 EXCEL非常普遍，其中 VBA程式易撰寫)，提供程式設計、航路規劃及相關研究人員參考。"" Navigational software often lack official standardization of the methods used and their accuracy due to commercial confidentiality. The “black box solutions” used by navigational systems are unknown, thus a logical and simple method to solve navigational problems must be presented. This project presents new navigational sailing formulas by the least squares method. For example, the traditional methods regarding the meridian arc formulas cannot be expressed as a closed form, they are often truncated to the first few terms for practical use and in doing so neglect the values not used, as traditional navigation only approximates the values, errors are neglected due to the inconvenient means to obtain the fast and convenient computer calculations in that current time and era. Traditional methods used Table look-ups to solve these problems, such as the meridian arc formula of Bowditch’s Table 7 to acquire one degree intervals of the meridian arc length. By forming an overdetermined system with known components of the traditional meridian arc equations and actual length of the meridian arc, the least squares method can be used to approximate the best fitting coefficients for the traditional meridian arc formulae and forms the new compact formulas. The new formulas are based on highly accurate values of the meridian arc for the WGS-84 ellipsoid datum, and are perfect for computational algorithms implemented in navigational software such as the Geographic Information Systems (GIS), Electronic Chart Display and Information Systems (ECDIS) and other Electronic Chart Systems (ECS), and in addition can be used to make much more accurate Tables for conventional uses, replacing those of neglected arguments in Bowditch’s Tables for navigational sailings. Accuracy of the Formulas obtained in this method is compared with other methods which simplify the arguments needed, in the already done new meridian arc formulas by the least squares method show that the new proposed formulas are shorter and accurate with negligible errors. The new formulas can be adapted to the accuracy needed and imply the needed number of coefficients. This can also shorten the calculations in navigation such as rhumb-line or great elliptic sailing on the ellipsoid because the meridian arc length is essential for these calculations. At the end of this study we wish to apply our results for open access in our website homepage so that other people from all around the world can learn what our study group has done, program codes for Mathematica® , Maple® and Matlab® will also be presented, as these tools are very useful in helping us apply least squares methods into navigational sailing, and are deemed a universal tool for algebra computer systems, the codes can provide a reference on just how we have applied them into our research, and finally, we hope to broaden to horizons of people that are interested in spheroid sailing and navigational programming."

Keyword(s)

最小平方法
代數符號運算
航海計算公式
Least Squares method
Navigational Sailing
Computer Algebra Systems

DSpace CRIS

Application of the Least Squares Method in Navigational Sailing

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