利用距離和距離變率進行自主式水下載具之觀測度分析

Abstract

本研究利用載具間距離與距離變率的量測所進行之觀測度分析，以探討自主式水下載具的定位問題。動態系統之觀測性可藉由對觀測矩陣之李導數（Lie derivatives）取格拉姆矩陣所獲得之格拉姆觀測矩陣進行研究。觀測度的高低可由格拉姆觀測矩陣的條件數之倒數（簡稱逆條件數）為衡量以量測資料估測水下載具位置之容易程度；當逆條件數提高，則系統觀測度將隨之改善。本研究提出一同時考慮距離與距離變率的條件數公式；與傳統僅使用距離量測相比較後發現，其觀測度隨 （相對速度向量與相對位置夾角）與 （相對位置與相對速度的比值）之變化趨勢一致，加入距離變率在夾角為 與 ，其系統觀測度有最佳改善。此外，以擴展卡爾曼濾波器（EKF）進行水下載具定位，其誤差協方差矩陣之跡數與逆條件數成反比關係。 本研究首先以模擬方式驗證所推導之公式，討論了直線、圓形、方型和螺旋路徑的逆條件數分析。再者，藉由2017年於基隆望海巷海域，以一配備有羅盤、都普勒速度記錄儀（DVL）與聲學層析儀之自主式水下載具所搜集之實驗數據進行分析。由於聲學層析儀發射以最大長度序列碼（m-sequences）的相位調變訊號，可同時量測距離和距離變率的資訊。實驗結果與模擬結果一致，加入距離變率的量測值可以增加逆條件數，因此提高載具定位的精準度。

This study investigates the localization of underwater vehicles via the observability analysis using both inter-vehicle range and range-rate measurements. An instantaneous observability of the dynamic system is defined by taking the Gramian matrix of the Lie derivatives, in which the condition number of the observability Gramian matrix is a metric of the observability. Better observability is obtained when the condition number is reduced (the inverse of the condition number is increased). The condition number formula is derived with not only the range but also the range-rate measurements. The overall characteristics of the inverse of the condition number for including both measurements are similar to those for using only range measurements. With additional range-rate measurements, the improvement is observed when the angle between the relative velocity vector and the position vector is close to or . With increasing the inverse of the condition number, the trace of the Extended Kalman Filter (EKF) error covariance matrix is reduced. The derived framework was demonstrated first using numerical simulations; Several routes including straight, circular, spiral and square paths were considered. Then a field experiment was conducted in WangHiXiang Bay, in 2017, with an Autonomous Underwater Vehicle (AUV) equipped with a compass, a tomographic sensor and a Doppler Velocity Log (DVL). The tomographic sensor transmits m-sequence signals, providing range and range-rate data simultaneously. The experiment results are consistent with the simulation results. Incorporating range-rate measurements improves the inverse of the condition number and therefore the localization of AUV.