基于自適應(yīng)有限沖激響應(yīng)-卡爾曼濾波算法的GPS/INS導(dǎo)航

靳 標(biāo)，李建行，朱德寬，郭 交※，蘇寶峰

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基于自適應(yīng)有限沖激響應(yīng)-卡爾曼濾波算法的GPS/INS導(dǎo)航

靳 標(biāo)1,2,3，李建行1，朱德寬1，郭 交1,2,3※，蘇寶峰1,2,3

（1. 西北農(nóng)林科技大學(xué)機(jī)電學(xué)院，楊凌 712100；2. 農(nóng)業(yè)農(nóng)村部農(nóng)業(yè)物聯(lián)網(wǎng)重點(diǎn)實(shí)驗(yàn)室，楊凌 712100； 3. 陜西省農(nóng)業(yè)信息感知與智能服務(wù)重點(diǎn)實(shí)驗(yàn)室，楊凌 712100）

導(dǎo)航定位系統(tǒng)一般采用卡爾曼濾波算法提高定位精度。傳統(tǒng)卡爾曼濾波算法的性能很大程度上依賴觀測(cè)噪聲的先驗(yàn)統(tǒng)計(jì)信息，不精確的統(tǒng)計(jì)特性將會(huì)降低定位精度。針對(duì)此問題，該文提出一種基于FIR（finite impulse response）預(yù)測(cè)模型的卡爾曼濾波算法。將FIR預(yù)測(cè)模型與卡爾曼濾波結(jié)合，F(xiàn)IR預(yù)測(cè)模型的系數(shù)可以通過求解一個(gè)凸二次規(guī)劃問題得到。該凸二次規(guī)劃以目標(biāo)的多項(xiàng)式運(yùn)動(dòng)規(guī)律為約束條件，以最小白噪聲增益為目標(biāo)函數(shù)，具有閉式解。仿真試驗(yàn)和實(shí)測(cè)結(jié)果均表明，在相同的參數(shù)設(shè)置條件下，基于FIR預(yù)測(cè)模型的卡爾曼濾波算法比傳統(tǒng)的卡爾曼濾波算法具有更高的估計(jì)精度，仿真結(jié)果表明定位精度提高29.54%，實(shí)測(cè)結(jié)果表明方向定位精度提高21.71%，方向定位精度提高22.62%。該算法可應(yīng)用于GPS接收信號(hào)的降噪處理，提高目標(biāo)狀態(tài)的定位精度。

導(dǎo)航；模型；FIR預(yù)測(cè)模型；自適應(yīng)卡爾曼濾波；全球定位系統(tǒng)

0 引 言

全球定位系統(tǒng)（global positioning system, GPS）和慣性導(dǎo)航系統(tǒng)（inertial navigation system, INS）在目標(biāo)定位、車輛導(dǎo)航和精準(zhǔn)農(nóng)業(yè)等領(lǐng)域得到了廣泛的應(yīng)用。然而由于衛(wèi)星信號(hào)遮擋、多路徑效應(yīng)和觀測(cè)誤差等因素的影響，濾波結(jié)果通常會(huì)出現(xiàn)較大誤差[1-2]。

卡爾曼濾波是GPS數(shù)據(jù)處理中常用的濾波算法[3-8]。其性能取決于狀態(tài)向量的動(dòng)態(tài)模型和描述噪聲特性的隨機(jī)模型[9-10]。目前相對(duì)應(yīng)的自適應(yīng)卡爾曼濾波算法有2種：一種是基于多模型的自適應(yīng)估計(jì)（multiple-model-based adaptive estimation, MMAE）[11-13]，另一種是基于新息的自適應(yīng)估計(jì)（innovation-based adaptive estimation, IAE）[14-15]。第一種方法是在不同的動(dòng)態(tài)模型和統(tǒng)計(jì)信息下，利用一組并行運(yùn)行的卡爾曼濾波器，將具有非零模型概率的所有模型結(jié)合起來。第二種方法是基于新息序列的變化直接通過計(jì)算觀測(cè)噪聲或過程噪聲協(xié)方差矩陣來完成自適應(yīng)濾波。MMAE和IAE方法中通常采用離散時(shí)間微分模型，例如用恒定速度的CV（constant velocity）模型和恒定加速度的CA（constant acceleration）模型來描述狀態(tài)變量的變化過程[16]。然而，位置、速度和姿態(tài)等狀態(tài)向量是相關(guān)的，準(zhǔn)確描述這些狀態(tài)的統(tǒng)計(jì)關(guān)系是非常困難的，當(dāng)先驗(yàn)信息不充分時(shí)，各個(gè)濾波狀態(tài)的耦合效應(yīng)也將給定位結(jié)果造成較大誤差。基于離散時(shí)間微分模型的卡爾曼濾波算法的另一個(gè)缺點(diǎn)是它高度依賴過程噪聲和觀測(cè)噪聲的先驗(yàn)統(tǒng)計(jì)信息。一般來說，過程噪聲和觀測(cè)噪聲的先驗(yàn)統(tǒng)計(jì)信息取決于運(yùn)動(dòng)過程和應(yīng)用場(chǎng)景，一般較難準(zhǔn)確獲得。濾波器的先驗(yàn)統(tǒng)計(jì)信息不足，會(huì)降低濾波狀態(tài)的估計(jì)精度，甚至導(dǎo)致濾波器估計(jì)結(jié)果出現(xiàn)發(fā)散現(xiàn)象。

自適應(yīng)卡爾曼濾波算法的研究主要集中在在線計(jì)算過程噪聲或觀測(cè)噪聲的協(xié)方差，目前很少有關(guān)于自適應(yīng)狀態(tài)模型的研究報(bào)道[17-18]。一個(gè)有效的狀態(tài)模型將在很大程度上有利于從觀測(cè)數(shù)據(jù)中提取有用的狀態(tài)信息。針對(duì)以上問題，本文針對(duì)白噪聲背景提出一種基于FIR預(yù)測(cè)模型的卡爾曼濾波算法。將FIR預(yù)測(cè)模型嵌入卡爾曼濾波算法，F(xiàn)IR預(yù)測(cè)模型的系數(shù)可以通過求解一個(gè)凸二次規(guī)劃問題得到。該凸二次規(guī)劃以目標(biāo)的多項(xiàng)式運(yùn)動(dòng)規(guī)律為約束條件，以最小白噪聲增益為目標(biāo)函數(shù)，具有閉式解。仿真試驗(yàn)和實(shí)測(cè)結(jié)果均表明在相同的參數(shù)設(shè)置條件下，基于FIR預(yù)測(cè)模型的卡爾曼濾波算法比傳統(tǒng)的卡爾曼濾波算法具有更高的估計(jì)精度。該算法可應(yīng)用于GPS/INS系統(tǒng)中的信息融合前的單狀態(tài)估計(jì)，或適用于GPS接收機(jī)后處理過程中的降噪。

1 FIR預(yù)測(cè)模型

目前，卡爾曼濾波算法的運(yùn)動(dòng)狀態(tài)模型大多數(shù)都屬于離散時(shí)間微分模型，本質(zhì)上是多項(xiàng)式模型。多項(xiàng)式模型能準(zhǔn)確地刻畫目標(biāo)運(yùn)動(dòng)狀態(tài)的變化規(guī)律，并且可以提供目標(biāo)在不同時(shí)刻的位置、速度和加速度等信息。但是這類模型是固定的，不能隨著濾波的過程和觀測(cè)噪聲強(qiáng)度的變化而改變，這在某種程度上影響了算法的性能[19]。本文提出一種針對(duì)白噪聲背景的狀態(tài)模型——FIR預(yù)測(cè)模型。該模型不僅能刻畫目標(biāo)的運(yùn)動(dòng)狀態(tài)，還可以濾除白噪聲。

1.1 由多項(xiàng)式模型到FIR模型

將式（3）帶入式（2），可得：

式（5）兩端都是關(guān)于的級(jí)數(shù)和，并且形式相同，因此通項(xiàng)公式相等，即

當(dāng)=0時(shí)，由式（6）可得：

當(dāng)>0時(shí)，假設(shè)當(dāng)-1時(shí)，式（6）成立，即

由式（6）可得

根據(jù)二項(xiàng)式定理可得

將式（10）帶入式（9），可得

由式（8）可知，式（11）等號(hào)右邊第二項(xiàng)為0，即可得到

即式（8）成立。

式（7）和式（8）可以表示為矩陣形式

1.2 最小白噪聲增益條件下的FIR預(yù)測(cè)模型

實(shí)際情況中觀測(cè)信號(hào)都是混有噪聲的，理論分析時(shí)通常假定噪聲為白噪聲。為了使噪聲通過濾波器的增益最小，通常希望白噪聲的濾波器增益[22]最小。白噪聲的濾波器增益為：

結(jié)合約束條件，以最小白噪聲增益作為優(yōu)化準(zhǔn)則，可得FIR預(yù)測(cè)模型的目標(biāo)函數(shù)

目標(biāo)函數(shù)（12）屬于帶有線性約束的凸二次規(guī)劃問題[23-24]，這是一個(gè)標(biāo)準(zhǔn)的具有條件的最小二乘問題，通常應(yīng)用于測(cè)量或大地測(cè)量中非線性問題。定義目標(biāo)函數(shù)式（18）的拉格朗日方程為

其中為拉格朗日乘子向量。

拉格朗日方程的梯度為

令矩陣

由于矩陣為行滿秩，因此矩陣為非奇異，于是可以得到式（19）拉格朗日方程的唯一最小解

即

其中，最優(yōu)拉格朗日乘子向量

即式（25）是優(yōu)化問題的唯一全局最小值。

2 基于FIR預(yù)測(cè)模型的卡爾曼濾波算法

目標(biāo)狀態(tài)的觀測(cè)方程為

基于FIR預(yù)測(cè)模型的卡爾曼濾波算法具體流程為：

1）濾波器初始化：

給定，值，并且≥+2；

2）狀態(tài)轉(zhuǎn)移矩陣計(jì)算：

3）狀態(tài)預(yù)測(cè)：

4）狀態(tài)更新：

增益矩陣：

后驗(yàn)估計(jì)均值：

后驗(yàn)估計(jì)誤差協(xié)方差：

5）令1，返回步驟3）。

3 試驗(yàn)與結(jié)果分析

3.1 仿真試驗(yàn)

根據(jù)理論推導(dǎo)可知速度和初始距離對(duì)該模型無影響，因此假設(shè)目標(biāo)以=20 m/s的速度勻速運(yùn)動(dòng)，初始距離為=10 km。觀測(cè)噪聲的方差=100 m2，采樣間隔Δ= 1 s，總的觀測(cè)時(shí)間為100 s[19]。根據(jù)本文的理論推導(dǎo)，設(shè)置FIR預(yù)測(cè)模型系數(shù)的個(gè)數(shù)分別為=2，3；在觀測(cè)噪聲為白噪聲的情況下，F(xiàn)IR預(yù)測(cè)模型系數(shù)的個(gè)數(shù)也為=2，3。FIR預(yù)測(cè)模型的階數(shù)均為=1。FIR預(yù)測(cè)模型的狀態(tài)噪聲協(xié)方差由式（17）給出，為了與FIR預(yù)測(cè)模型保持一致，CV模型的狀態(tài)噪聲協(xié)方差為[15]

從圖1的仿真結(jié)果可以看出：

1）在白噪聲背景下FIR預(yù)測(cè)模型（=1，=2）的情況和CV模型有相同的定位精度（由于噪聲的隨機(jī)性引起圖1中CV模型與FIR預(yù)測(cè)模型（=1,=2）的2條線不完全重合，如果Monte-Carlo次數(shù)此時(shí)足夠多是可以重合的），證明了上文中提到的FIR預(yù)測(cè)模型與CV模型等價(jià)，并且都可以用來描述勻速運(yùn)動(dòng)。

2）在定位精度上，F(xiàn)IR預(yù)測(cè)模型（=1，=3）的情況優(yōu)于CV模型，其精度提高29.54%。因?yàn)樵撃Ｐ筒粌H可以滿足多項(xiàng)式運(yùn)動(dòng)的CV模型，而且具有一定的降噪效果。另外，在白噪聲背景下，考慮到使濾波的增益最小，在某種程度上進(jìn)一步降低了噪聲的影響。

3）比較圖1a和圖1b，可以看出最優(yōu)FIR預(yù)測(cè)模型在參數(shù)設(shè)置與實(shí)際運(yùn)動(dòng)不匹配時(shí)，要優(yōu)于傳統(tǒng)的離散時(shí)間微分模型。

注：qr代表狀態(tài)噪聲強(qiáng)度，N代表FIR預(yù)測(cè)模型的階數(shù)，M表示FIR預(yù)測(cè)模型系數(shù)的個(gè)數(shù)。

3.2 實(shí)測(cè)試驗(yàn)

為了評(píng)估所提出算法的定位精度，2018年8月10日在西北農(nóng)林科技大學(xué)北校區(qū)操場(chǎng)進(jìn)行多次實(shí)際測(cè)試。測(cè)試設(shè)備如圖2a所示，GNSS RTK系統(tǒng)由HI-TARGET測(cè)量?jī)x器有限公司制造，型號(hào)為A10，其定位精度為（10+ 1′10–6′）m，表示移動(dòng)站與基站之間的距離，實(shí)測(cè)時(shí)≤300 m。手持式GNSS接收機(jī)為集思寶G130型號(hào)。手持式GNSS接收機(jī)只接收GPS衛(wèi)星信號(hào)，理論定位精度為3～5 m。以GNSS RTK系統(tǒng)測(cè)得的軌跡作為真實(shí)軌跡的參考值，手持式GNSS接收機(jī)測(cè)得的數(shù)據(jù)作為需要濾波的軌跡。根據(jù)理論推導(dǎo)可知，F(xiàn)IR預(yù)測(cè)模型與運(yùn)動(dòng)速度無關(guān)，因此移動(dòng)站與手持式GNSS接收機(jī)由1個(gè)人手持以近似1 m/s勻速繞操場(chǎng)1周。

測(cè)試10次取平均值，將數(shù)據(jù)導(dǎo)入Google earth，測(cè)試軌跡如圖2b所示。與仿真試驗(yàn)的采樣間隔保持一致，2個(gè)接收機(jī)的采樣頻率均為1 Hz。

圖2 測(cè)試設(shè)備及測(cè)試軌跡

圖3 X方向和Y方向位置誤差

圖3a和圖3b分別給出了CV模型、白噪聲狀態(tài)背景下FIR預(yù)測(cè)模型（=1,=3）算法濾波后方向（東西方向）和方向（南北）位置誤差。由圖3可以看出，在白噪聲狀態(tài)背景下FIR預(yù)測(cè)模型（=1,=3）算法明顯優(yōu)于CV模型，經(jīng)計(jì)算東西方向定位精度提高21.71%，南北方向定位精度提高22.62%。因?yàn)樵趧蛩僦本€運(yùn)動(dòng)時(shí)FIR預(yù)測(cè)模型可以在最小均方誤差的意義下盡可能地降低白噪聲的影響，并且FIR預(yù)測(cè)模型可以通過新息序列利用在線信息。

4 結(jié) 論

基于FIR預(yù)測(cè)模型的卡爾曼濾波算法可以由求解凸二次規(guī)劃問題得到最優(yōu)解。白噪聲背景下的FIR預(yù)測(cè)模型不僅滿足狀態(tài)變量的多項(xiàng)式約束，而且降低了噪聲的影響。該算法可應(yīng)用于GPS/INS系統(tǒng)中的信息融合前的單狀態(tài)估計(jì)，或適用于GPS接收機(jī)后處理過程中的降噪。相比于傳統(tǒng)算法，該算法具有以下優(yōu)點(diǎn)：

1）基于FIR預(yù)測(cè)模型的濾波算法可認(rèn)為對(duì)測(cè)量數(shù)據(jù)進(jìn)行2次濾波，即首先由FIR濾波，然后用卡爾曼濾波器對(duì)濾波數(shù)據(jù)進(jìn)行濾波。因此，基于FIR預(yù)測(cè)模型的卡爾曼濾波算法的去噪效果優(yōu)于傳統(tǒng)的對(duì)數(shù)據(jù)進(jìn)行1次濾波的去噪效果。

2）基于FIR預(yù)測(cè)模型的卡爾曼濾波算法通過一定程度的多項(xiàng)式逼近運(yùn)動(dòng)軌跡，利用FIR預(yù)測(cè)模型預(yù)測(cè)位置。線性化的技術(shù)可以應(yīng)用到相似的的非線性估計(jì)中。

3）基于FIR預(yù)測(cè)模型的卡爾曼濾波算法的計(jì)算量與傳統(tǒng)卡爾曼濾波相當(dāng)，F(xiàn)IR預(yù)測(cè)模型的系數(shù)具有閉式解。該算法定位精度高于傳統(tǒng)的卡爾曼濾波算法，實(shí)測(cè)試驗(yàn)結(jié)果表明，東西方向定位精度提高21.71%，南北方向定位精度提高22.62%。

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GPS/INS navigation based on adaptive finite impulse response-Kalman filter algorithm

Jin Biao1,2,3, Li Jianxing1, Zhu Dekuan1, Guo Jiao1,2,3※, Su Baofeng1,2,3

(1.712100; 2.712100,; 3.712100,)

Global positioning system (GPS) and inertial navigation system (INS) are widely used in target positioning, vehicle navigation, precision agriculture and other fields. However, due to factors such as satellite signal occlusion, multi-path effect and observation error, the filtering results usually have large errors. Kalman filtering algorithm is generally used in navigation and positioning system to improve positioning accuracy. The performance of kalman filter algorithm depends on the dynamic model of state vector and the random model describing noise characteristics. There are 2 corresponding adaptive kalman filtering algorithms: one is the multiple-model-based adaptive estimation (MMAE); the other is the innovation-based adaptive estimation (IAE). The first method is to combine all models with non-zero model probability by using a set of parallel kalman filters under different dynamic models and statistical information. The second method complete the adaptive filtering directly by calculating the observation noise or process noise covariance matrix based on the change of information sequence. In MMAE and IAE methods, discrete time differential models are usually adopted, such as the constant velocity CV model and constant acceleration CA model, to describe the change process of state variables. However, the state vectors such as position, velocity and attitude are correlated, and it is very difficult to accurately describe the statistical relations of these states. When the prior information is not sufficient, the coupling effect of each filtering state will also cause large errors to the positioning results. Another disadvantage of kalman filter algorithm based on discrete time differential model is that it highly depends on the prior statistical information of process noise and observation noise. Generally, the prior statistical information of process noise and measurement noise depends on the motion process and application scene, which is difficult to be obtained accurately. Insufficient prior statistical information of the filter will reduce the estimation accuracy of the filter state and even lead to the divergence of the filter estimation results. The research of adaptive kalman filter algorithm is mainly focused on the covariance of online calculation process noise or measured noise. In order to improve the accuracy of navigation and positioning, an adaptive kalman filter algorithm based on FIR (finite impulse response) prediction model for white noise background was proposed in this paper. Firstly, the continuous trajectory function of moving target was approximated by an-order polynomial with arbitrary precision, and the FIR prediction model polynomial was obtained. The FIR prediction model coefficient was obtained by solving a convex quadratic programming problem, and the optimal solution of FIR prediction model coefficient was solved by lagrange multiplier method. The convex quadratic programming taken the polynomial motion law of the target as the constraint condition and the minimum white noise gain as the objective function, and the optimal solution could not only satisfy the constraints of the target's motion state, but also had the effect of de-noising to a certain extent. Finally, the proposed FIR prediction model was combined with kalman filter. Simulation test and the measurement results showed that kalman filtering algorithm based on FIR prediction model had higher estimation accuracy than the traditional kalman filtering algorithm under the same parameter settings, and the simulation experiment results showed that the localization precision was increased by 29.54%, the measured experimental results showed that positioning accuracy ineast-west direction increased by 21.71%, positioning error innorth-south direction increased by 22.62%. The proposed algorithm could be used for single state estimation before information fusion in loosely coupled GPS/INS, and also for noise reduction in post-processing of GPS receivers.

navigation; models; FIR prediction model; adaptive Kalman filtering; global positioning system

靳 標(biāo)，李建行，朱德寬，郭 交，蘇寶峰. 基于自適應(yīng)有限沖激響應(yīng)-卡爾曼濾波算法的GPS/INS導(dǎo)航[J]. 農(nóng)業(yè)工程學(xué)報(bào)，2019，35(3)：75－81. doi：10.11975/j.issn.1002-6819.2019.03.010 http://www.tcsae.org

Jin Biao, Li Jianxing, Zhu Dekuan, Guo Jiao, Su Baofeng. GPS/INS navigation based on adaptive finite impulse response-Kalman filter algorithm[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2019, 35(3): 75－81. (in Chinese with English abstract) doi：10.11975/j.issn.1002-6819.2019.03.010 http://www.tcsae.org

2018-09-11

2019-01-10

國(guó)家自然科學(xué)基金（61701416）；中央高?；究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金（2452017127）；農(nóng)業(yè)農(nóng)村部農(nóng)業(yè)物聯(lián)網(wǎng)重點(diǎn)實(shí)驗(yàn)室開放基金課題（2017AIOT-06）

靳 標(biāo)，講師，博士，研究方向?yàn)镚PS信號(hào)處理和精準(zhǔn)農(nóng)業(yè)。Email：biaojin@nwafu.edu.cn

郭 交，副教授，博士，研究方向?yàn)镚PS信號(hào)處理和精準(zhǔn)農(nóng)業(yè)。Email：jiao.g@163.com

10.11975/j.issn.1002-6819.2019.03.010

TP391.4

A

1002-6819(2019)-03-0075-07