对偶四元数导航算法及非线性高斯滤波研究

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1、附件6作者姓名：武元新论文题目：对偶四元数导航算法与非线性高斯滤波研究作者简介：武元新，男，1976年11月出生，2000年04月师从于国防科学技术大学胡德文教授，于2005年12月获博士学位。中 文 摘 要本论文深入研究了导航系统中所涉及的两个基本理论问题：导航信息的数学描述和数值解算；导航信息的最优实时融合（即滤波）策略。目前，导航、机器人和计算机视觉等研究领域普遍以向量代数为工具研究空间运动学问题。但是当研究一般性刚体运动时，向量代数中没有数学工具可以对转动和平移统一描述。幸而，从运动学的观点来看，一般性刚体运动可以拆分为两个子运动：定点转动和平移，其中转动完全独立于平移。这意味着我们可

2、以转而采用另一种方式处理转动和平移。在导航领域中，方向余弦矩阵DCM或四元数用来描述转动，向量用来描述平移。捷联惯性导航算法亦是如此，即针对姿态积分和速度/位置积分，必须分别设计不同的算法来实现。最近的研究发现圆锥算法和划船算法之间存在对偶性或等价性，也就是说，可以通过一个简单的数学公式把圆锥算法变换成相应的划船算法，但是导航算法的设计和实现仍然很棘手。惯性导航本质上要解决的是一个三维空间的刚体运动学问题。存在统一、简洁地描述一般性刚体运动的数学语言吗？如果存在，可否利用该数学工具设计思路明晰、复杂度低的导航算法？这两个问题的答案都是肯定的。论文的前半部分在建立基于对偶四元数的捷联惯性导航系统

3、理论方面作了一些探索性工作。1. 作为几何代数的子集，对偶四元数是刻画一般性刚体运动的最简洁、最有效的数学工具，可以用来研究包括惯性导航在内的所有刚体运动学（和动力学）问题。根据Bar-Itzhack分离坐标系的思想，论文第二章运用对偶四元数代数重新诠释捷联式惯性导航的基本原理，得到了三个对偶四元数运动学方程，其形式均与传统算法中的姿态四元数微分方程一致。借鉴成熟的姿态四元数积分的双速算法结构，设计了一个数值积分算法求解以上三个运动学方程，构建了基于对偶四元数的捷联惯性导航算法。对偶四元数算法将传统算法中的圆锥、划船和卷轴修正整合到一起，大大简化了算法结构，降低了实现难度。从多个侧面对对偶四元

4、数算法和传统算法进行了理论分析和比较，证实了螺旋运动实质上是圆锥运动和划船运动的复合运动，揭示了传统算法中圆锥算法和划船算法之间存在对偶性/等价性的根本原因，导出了对偶四元数算法和传统算法误差的解析表达式，从理论上证明了在高精度和高动态环境中，对偶四元数算法的精度将优于传统算法。设计并实现了理想情况下以及考虑不同级别惯性器件误差的实际情况下的仿真试验，数值结果为理论分析提供了强有力的佐证。对高精度导航系统和大机动场景来说，对偶四元数算法是一个更好的选择。最后，提供了根据惯性器件配置和传统算法的转折频率选择合适导航算法的指导原则。不久的将来，对偶四元数算法有望在基于超冷原子干涉仪的精确惯性导航系

5、统中发挥重要作用。（已发表于Strapdown inertial navigation system algorithms based on dual quaternions, IEEE Trans. on Aerospace and Electronic Systems, 2005；On A unified mathematical framework for strapdown algorithm design, Journal of Guidance, Control, and Dynamics, 2006; Observability Analysis of Rotation Esti

6、mation by Fusing Inertial and Line-Based Visual Information: A Revisit, Automatica, 2006）最近，Ohio大学的Soloviev博士在频域而不是时域中实现了传统导航算法。据称，该频域方法在降低由圆锥运动和划船运动引起的不可交换性误差的能力方面有很好的改善。在频域中实现对偶四元数导航算法将是一项非常有意义的工作。（2006年，此后续工作获得国家自然科学基金青年科学基金资助,“频域中的对偶四元数捷联惯性导航算法研究”，60604011）。2. 在基于对偶四元数的捷联惯性导航理论框架中，姿态、速度和位置等所有的导航

7、参数都可以从三个对偶四元数运动学方程的解中直接导出，这使得完全以四元数代数建立误差传递模型成为可能。论文第三章运用对偶四元数代数研究了捷联式惯性导航的误差特性。导出了两个完全用四元数代数表达的误差模型：加性对偶四元数误差模型和乘性对偶四元数误差模型。这两个误差模型可用来搭建基于对偶四元数的组合导航滤波器。（已发表于Strapdown inertial navigation system using dual quaternions: error analysis, IEEE Trans. on Aerospace and Electronic Systems, 2006）3. 对偶四元数导航算

8、法直接输出的导航参数是在地球坐标系中表达的，但是在GPS导航和测地学等应用中，经常需要在当地地理坐标系中表达导航参数。论文第四章研究了对偶四元数导航算法中所涉及的一个子问题从地球坐标系到当地地理坐标系的坐标变换问题。提出了一个基于迭代Newton-Raphson方法的坐标变换快速算法。除了靠近地心的一个小区域之外，新算法不存在奇异和不收敛的情况。与已有算法的分析和比较显示，新算法精度更高、计算量更小。（已发表于Algorithm of Earth-centered Earth-fixed coordinates to geodetic coordinates, IEEE Trans. on A

9、erospace and Electronic Systems, 2003）作为一种完全自主的航迹推算方法，惯性导航存在一个固有缺陷，即其误差随着时间累计。为了克服这一缺陷，必须依靠非惯性传感器引入外部信息反馈，与惯性导航系统共同组成一个稳定的闭环系统，抑制惯性导航误差的发散。多种多样的组合导航方案应运而生。组合导航是利用多种传感器提供的互补信息来提高导航系统的精度和冗余度的技术。无论采用何种配置（惯性导航、卫星导航、雷达、摄像机、多普勒测速仪、高度计等传感器间的优化组合）构成组合导航系统，都需要选择一个最优的在线信息融合策略将多种传感器提供的信息进行有效融合这正是非线性滤波器在组合导航系统中

10、所起的作用，即非线性滤波是组合导航系统中信息融合的基石。事实上，目前广泛应用的扩展Kalman滤波（Extended Kalman Filter, EKF）本身就是在组合导航系统需求牵引下的研究成果。自从20世纪60年代在阿波罗计划中首次实现以来，EKF已经在工程界盛行了近半个世纪，并在事实上成为工程界的一个标准组件。同时，EKF是一种近似非线性高斯滤波器，几十年来的工程经验显示其自身仍存在很多问题，比如需要将待处理的非线性系统模型进行线性化，滤波过程中容易发散等等。近年来，非线性滤波研究取得了众多的成果，组合导航系统的信息融合策略也逐渐开始考虑采用更先进的非线性滤波器。那么，可否找到或设计出

11、替代EKF的通用非线性滤波器？从信息论的角度讲，这是一个寻找比EKF更优越的实时信息融合策略（即滤波）的问题。论文的后半部分着重研究了非线性高斯滤波问题。4. Bayes推演为动态系统的状态估计问题提供了最优的解决方案，但是由于其最优解需要传播整个概率密度，所以Bayes推演通常是没有解析解的，其求解必须使用近似方法，比如假设概率密度近似服从高斯或混合高斯分布。在这一假设下设计出来的滤波器常被称作高斯滤波器。目前存在多种高斯滤波器，其背景也迥然不同。面对众多的高斯滤波器，如何确定哪一种最适合用来解决手上的滤波问题？论文第五章国际上首次提出从多维数值积分的观点出发对高斯滤波器进行研究，在统一的数

12、值积分框架下导出了文献中的近似高斯滤波器。这些高斯滤波器都是运用某种数值积分方法（如Gauss-Hermite积分公式，单项式精确公式和函数拟合方法）对一般形式的高斯滤波器的近似。基于多维数值积分的观点，应用精度、效率和稳定因子等指标对现有高斯滤波器进行了诠释，并对各种高斯滤波器的精度进行了排序。数值仿真结果与理论分析吻合得很好。数值积分观为工程上选择合适的近似高斯滤波器奠定了基础，对运用更好的积分方法设计高效、稳定的滤波器具有一定的指导意义。（已发表于Comments on Gaussian particle filtering, IEEE Trans. on Signal Processi

13、ng, 2005；A numerical-integration perspective on Gaussian filters, IEEE Trans. on Signal Processing, 2006）5. 论文第六章研究了UKF（Unscented Kalman filter）滤波器的两种实现：扩展UKF和非扩展UKF。它们都可以用于具有加性噪声的非线性动态系统。此前普遍认为：在这种特殊但却常见的情况下，使用非扩展形式的UKF可以降低计算复杂度，并且不会降低滤波精度。本章旨在证实这种看法是不正确的，相反，在加性系统和观测噪声的情况下使用非扩展形式的UKF很可能会损失滤波精度。本章首先

14、导出了扩展UT与非扩展UT等价的前提条件，进而指出扩展UKF和非扩展UKF之间的根本区别在于前者只需要在一次滤波迭代过程中产生一次sigma点集，而后者则产生两次sigma点集，即必须重新产生一个新的sigma点集以纳入加性系统噪声的影响。这个区别通常有利于扩展UKF，因为奇次矩被变换后的sigma点集捕获，并得以在单个滤波迭代过程内传播。另外，如果有意重新产生一个新的但却不必要的sigma点集，扩展UKF将等价于非扩展UKF，两者的滤波结果也将完全相同。最后，考察了信号处理领域的一个典型例子，仿真实例的结果与分析结论一致。（已发表于Unscented Kalman filtering for

15、 additive noise case: augmented versus non-augmented, IEEE Signal Processing Letters, 2005; Comments on “Performance evaluation of UKF-based nonlinear filtering”, Automatica, 2007）6. 最近提出的高斯粒子滤波器（Gaussian Particle Filtering, GPF）是一个基于Bayes采样思想的高斯滤波器。它通过Monte Carlo积分和Bayes更新规则对传统的高斯滤波器进行了推广。论文第七章从两个不

16、同的角度出发对GPF进行拓展。首先，提出了一个半高斯滤波器（QuasiGPF）。QuasiGPF容许先验概率密度为非高斯的，具有比GPF更高的理论精度。仿真研究表明QuasiGPF的精度确实比GPF有很大的提高。理论上，验后概率密度可以假定为容易采样的任意分布，比如混合高斯。这种情况下，QuasiGPF可以用来取代GPF以构建更高精度的混合高斯粒子滤波器。低偏差序列的目的是生成确定性的相关样本，并使这些样本在目标空间中尽量均匀的分布。使用低偏差序列的准Monte Carlo方法可用于近似计算多维积分或序贯Bayes概率推演。我们基于准Monte Carlo方法研究了一类特殊的Bayes滤波GP

17、F。数值结果表明基于准Monte Carlo方法的GPF比使用随机数的传统GPF具有更小的MSE和更快的收敛速度。关键词：惯性导航, 对偶四元数, 数值积分, 误差特性, 坐标变换, 非线性高斯滤波, 粒子滤波, 低偏差序列Dual-Quaternion Navigation Algorithm and Nonlinear Gaussian FilteringWu YuanxinABSTRACTThe thesis has investigated two theoretical problems involved in navigation systems. One is the mathe

18、matical representation and computation of navigation information; the other is the optimal real-time fusion strategy, i.e., filtering, of navigation information. So far in most, if not all, fields, researches have been made within the framework of vector algebra. When it comes to the general displac

19、ement of a rigid body, however, there is no such a mathematical tool in vector algebra as to treat rotation and translation simultaneously. Fortunately, from a viewpoint of kinematics the general displacement can be taken apart into two separate motions, i.e. fixed-point rotation and translation, in

20、 which rotation is completely independent of translation. This means that we can otherwise treat rotation and translation in a different manner. In the navigation community, DCM/quaternion and vector are chosen to represent rotation and translation, respectively. So are the strapdown INS algorithms.

21、 Individual algorithm as mentioned above has to be structured for attitude integration and velocity/position integration, respectively. The characteristics of duality/equivalence between the coning and sculling algorithms were revealed recently, but the algorithm design and implementation are still

22、rather involved.In essence, inertial navigation is to solve the kinematic problem of a three-dimensional rigid body. Is there any mathematical language that represents rotation and translation in a consolidating and compact manner? Is it possible to reduce the perplexing and error-prone strapdown al

23、gorithms to some extent? The answers to these questions are both affirmative. The first half of this thesis has done some pilot works on the strapdown inertial navigation theory founded on dual quaternion.1. As a subset of geometry algebra, dual quaternion is the most concise and efficient mathemati

24、cal tool to represent the general rigid motion. It can be used to address all rigid kinematic (and dynamic) problems, including inertial navigation of course. Benefiting from Bar-Itzhacks split-coordinate scheme, Chapter 2 reinterprets the rationale of the strapdown inertial navigation in terms of d

25、ual quaternion algebra, obtaining three dual quaternion kinematic equations that take the same forms as the conventional attitude quaternion rate equation. Borrowing the traditional two-speed approach originally developed in conventional attitude integration, we design one new numerical integration

26、algorithm to solve the three kinematic equations, thus obtaining the dual quaternion navigation algorithm, which integrates the coning, sculling and scrolling corrections all together, considerably simplifying the algorithm structure and implementation complexity. The new navigation algorithm is ana

27、lyzed and compared with the conventional one from various aspects. It is shown that screw motion in itself consists of coning motion and sculling motion. The duality between the coning and sculling corrections, raised in the recent literature, is fundamentally explained. The superiority of the new a

28、lgorithm in accuracy is analytically derived. A variety of simulations are carried out to support the analytic conclusions, including those with ideal inertial sensors and those with non-ideal ones. The numerical results agree well with the analyses. The new algorithm turns out to be a better choice

29、 than the conventional algorithm for high-precision navigation systems and high-maneuver applications. Several guidelines in choosing a suitable navigation algorithm are also provided according to the inertial sensors configuration and the turning frequency of the conventional algorithm. In the near

30、 future, the dual quaternion algorithm is expected to take an important role in ultra-cold atom interferometry based precision inertial navigation systems. (Published in Strapdown inertial navigation system algorithms based on dual quaternions, IEEE Trans. on Aerospace and Electronic Systems, 2005；O

31、n A unified mathematical framework for strapdown algorithm design, Journal of Guidance, Control, and Dynamics, 2006; Observability Analysis of Rotation Estimation by Fusing Inertial and Line-Based Visual Information: A Revisit, Automatica, 2006)Recently, Dr. Soloviev in Ohio University has implement

32、ed the conventional strapdown algorithm in frequency domain rather than in the customary time domain. It is claimed that the frequency-domain approach has a significant improvement in the ability to reduce the noncommutativity errors incurred by the coning and sculling motion. Redesigning the dual q

33、uaternion-based algorithm in the frequency domain would be a significant work. (In 2006, the proposal obtained the support of NSFC, entitled “Dual Quaternion Strapdown Inertial Navigation Algorithm in Frequency Domain”, 60604011)2. Within the dual-quaternion mechanism, all navigation quantities (inc

34、luding attitude, velocity and position) can be derived by manipulating the solutions to the three kinematic equations. This makes it possible to model the error propagation completely in terms of quaternion algebra. Chapter 3 is devoted to error characteristics of the strapdown inertial navigation u

35、sing dual quaternion. Two new error models in terms of quaternion algebra are developed: the additive dual quaternion error model and multiplicative dual quaternion error model. Both are expected to facilitate the future dual quaternion-based integrated navigation filter. (Published in Strapdown ine

36、rtial navigation system using dual quaternions: error analysis, IEEE Trans. on Aerospace and Electronic Systems, 2006)3. Dual quaternion navigation algorithm directly outputs navigation parameters in the Earth frame, but in applications such as GPS navigation and geodesy, we often need them expresse

37、d in the local level frame. Chapter 4 investigates a sub-problem involved in the dual quaternion navigation algorithm, i.e., the transformation from Earth-centered Earth-fixed coordinates to geodetic coordinates. We come up with an iterative approach using the Newton-Raphson method that has good eff

38、iciency and accuracy and is free from singularity and divergence except in a small region near the center of the Earth. Comparisons with existing methods show the new algorithm has much higher accuracy and lower arithmetic complexity. (Published in Algorithm of Earth-centered Earth-fixed coordinates

39、 to geodetic coordinates, IEEE Trans. on Aerospace and Electronic Systems, 2003)As a self-contained dead reckoning method, inertial navigation has an inherent limitation with its error accumulating as time goes. So inertial navigation must be integrated with extraneous information feedback to form a

40、 stable and closed-form integrated navigation system, so as to limit the error accumulation. Many kinds of integrated schemes emerge as the times require. Integrated navigation is aimed to improve the system accuracy and redundancy using compensatory information from various sensors. Whichever confi

41、guration (inertial navigation, satellite navigation, radar, camera, Doppler speedometer and altimeter, etc.) the integration takes, we must rely on an optimal on-line information fusing strategy, i.e., filtering, to efficiently integrate various information sources. Therefore, nonlinear filtering is

42、 the cornerstone of any integrated navigation system. In fact, the well-known EKF is one direct outcome of integrated navigation system requirements. Since its first application in Apollo Project in 1960s, EKF has prevailed for half a century and actually becomes a standard component in engineering.

43、 On the other hand, EKF is a kind of approximate Gaussian filter and experiences has indicated its limitations, e.g., linearization of any considered nonlinear system and being apt to divergence. In recent decades, nonlinear filtering researches have made many achievements and advanced nonlinear fil

44、ters are being considered as a feasible information fusion strategy for integrated navigation.Then, could we find or design general nonlinear filters to replace EKF? From the aspect of information theory, it belongs to a problem of searching real information fusion strategies, i.e., filtering, super

45、ior to EKF.Specifically, the latter part of the thesis has been focused on nonlinear Gaussian filtering.4. Bayesian inference provides an optimal solution framework for dynamic state estimation problems. Because the Bayesian solution requires the propagation of the full probability density, in gener

46、al the optimal nonlinear filtering is analytically intractable. Approximations are thus necessary, e.g., Gaussian approximation to the posterior probability density. The class of filters derived under Gaussian assumption is commonly called as the Gaussian filters. So far, there has been a variation

47、of Gaussian filters that derived themselves from very different backgrounds. A question now arises: with so many different Gaussian filters, how to decide which one is suitable for a filtering problem in hand? Chapter 4 reviews the state of art of Gaussian filters from the perspective of numerical i

48、ntegration. Specifically, we present in a unified numerical-integration framework the derivation of a number of approximate Gaussian filters. It shows that all Gaussian filters are approximations of the general Gaussian filter by using a specific numerical integration method of some kind or another,

49、 such as the Gauss-Hermite product rule, rules exact for monomials and methods of approximation. This perspective provides a well-founded understanding of all the existing Gaussian filters with respect to accuracy, efficiency and stability factor. The analytical findings are tabulated, from which a

50、ranking of accuracy of various Gaussian filters is derived. The numerical results agree nicely with the analytical ranking list. We believe that this perspective has set a good foundation for selection of Gaussian filters in practice and hopefully be useful to design more efficient and stable filter

51、s by employing better numerical integration methods. (Published in Comments on Gaussian particle filtering, IEEE Trans. on Signal Processing, 2005；A numerical-integration perspective on Gaussian filters, IEEE Trans. on Signal Processing, 2006)5. Chapter 6 analyzes and compares two alternative versio

52、ns of unscented transformation (UT)-based filters for the nonlinear dynamic system with additive noises: the non-augmented unscented Kalman filter (UKF) and the augmented UKF. They can be both applied to nonlinear dynamic systems with additive noises. It is now believed that for the special (but oft

53、en found) case where process and measurement noises are additive, the computational complexity can be reduced by using the non-augmented form, which presumably yields similar results, if not the same. In this chapter we will show that this assumption is not quite correct and that the non-augmented U

54、KF usage can lead to noticeable losses in accuracy. Firstly, it is proved that the non-augmented UT is identical to the augmented counterpart only if certain condition is satisfied. We point out that the basic difference between the augmented and non-augmented UKFs is that the former draws sigma poi

55、nts only once in a recursion while the latter has to redraw a new set of sigma points to incorporate the effect of additive process noise. This difference generally favors the augmented UKF in that the odd-order moment information is captured by the transformed sigma points and well propagated withi

56、n one recursion. On the other hand, if a new (but unnecessary) set of sigma points were redrawn in the augmented UKF, it would be identical to and yield exactly the same results as the non-augmented UKF. The simulation results of a representative example agree well with our conclusions. (Published i

57、n Unscented Kalman filtering for additive noise case: augmented versus non-augmented, IEEE Signal Processing Letters, 2005; Comments on “Performance evaluation of UKF-based nonlinear filtering”, Automatica, 2007)6. Gaussian particle filter (GPF) is a kind of Gaussian filter based on Bayesian samplin

58、g. It actually extends the conventional Gaussian filter using Monte Carlo integration and the Bayesian update rule. Chapter 7 further extends GPF from two different aspects. Firstly, a so-called quasi-Gaussian particle filter (QuasiGPF) is proposed that generalizes the GPF by relaxing the Gaussian r

59、estriction on the prior probability density function. Considering the non-Gaussianity of the prior probability, the QuasiGPF is provably superior to the GPF. Numerical results show its remarkably improved performance over the GPF. Theoretically, the posterior probability could be assumed to be any o

60、ther distribution as long as it was readily sampled, e.g., mixed Gaussian. In such a case, it is promising for the QuasiGPF to be used to construct more superior filter than the GPF-based Gaussian sum particle filter. Secondly, the low discrepancy sequences are invented to distribute deterministic c

61、orrelated draws over the target domain as uniformly as possible. The quasi-Monte Carlo method using these sequences can be used to approximate multidimensional integrals and further to sequential Bayesian statistical inference. We investigate a special version of Bayesian filtering, i.e., the GPF, v

62、ia quasi-Monte Carlo method. Numerical results show that the new GPF outperforms the conventional GPF using random numbers in the sense of having lower MSE and faster convergence.Keywords: inertial navigation, dual quaternion, numerical integration, error characteristics, coordinate transformation, nonlinear Gaussian filtering, particle filtering, low-discrepancy sequence9 / 9文档可自由编辑打印