误差状态卡尔曼滤波器中的四元数动力学

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本文档深入探讨了四元数在误差状态卡尔曼滤波器中的应用，主要针对的是四元数的定义、性质及其在旋转表示和运动估计中的作用。四元数是一种复数扩展，最初由威廉·罗恩德于19世纪初引入，用于处理三维空间中的旋转问题，因其简洁性和高效性而在许多工程领域，尤其是计算机图形学、姿态控制和导航系统中得到广泛应用。章节一首先介绍了四元数的基本概念。1.1节阐述了四元数的定义，它由实部和三个虚部组成，可以看作是复数的扩展，提供了在三维空间中进行旋转的一种更直观的方式。此外，还讨论了不同四元数表示形式，如标准形式、纯四元数和单位四元数。 1.2节详述了四元数的主要性质，包括加法（满足交换律但不满足结合律）、乘法（非交换但满足分配律）、单位元、共轭（用于求逆和计算长度）、以及四元数的逆和单位化操作。此外，还涉及到了纯四元数和一般四元数的幂运算、指数函数和对数函数的表示，这些都是在处理旋转和运动学问题时必不可少的工具。章节二进一步探讨了四元数与旋转的关系。2.1节展示了三维向量的旋转公式，它是基于四元数乘法实现的。2.2节介绍了旋转群SO(3)，这是所有旋转构成的群，其元素对应于保持向量长度不变的旋转。2.3节着重于旋转群与旋转矩阵之间的联系，通过指数映射和资本化指数映射将四元数转换为旋转矩阵，这是实际应用中常见的操作。特别是2.3.3中提到的罗德里格斯公式，它将旋转矩阵与旋转向量关联起来，是将四元数转换为欧拉角或其他参数形式的关键步骤。这一部分对于理解四元数在误差状态卡尔曼滤波器中的旋转估计和滤波至关重要。本文提供了一个坚实的理论基础，帮助读者理解如何利用四元数的数学特性来有效地处理和估计旋转和运动误差，特别是在高精度定位和控制系统中，误差状态卡尔曼滤波器结合四元数方法能够提高数据融合的准确性和效率。对于从事航天、机器人技术或任何涉及三维空间旋转的工程师和研究者来说，这篇文章是一份宝贵的参考资料。

show that both representations are equally valid. They exhibit a lot of similarities, both

conceptually and algebraically, as the reader will appreciate in Table 1. Perhaps, the

most important diﬀerence is that the unit quaternion group constitutes a double cover of

SO(3) (thus technically not being SO(3) itself), something that is not critical in most of

our applications.

The table is inserted upfront for the sake of a rapid comparison and

evaluation. The rotation matrix and quaternion representations of SO(3) are explored in

the following sections.

2.3 The rotation group and the rotation matrix

The operator r() is linear, since it is deﬁned from the scalar and vector products, which

are linear. It can therefore be represented by a matrix R ∈ R

3×3

, which produces rotations

to vectors v ∈ R

through the matrix product,

r(v) = R v . (58)

Injecting it in (55a), using the dot product ha, bi = a

b and developing we have that for

all v,

(Rv)

(Rv) = v

Rv = v

v , (59)

yielding the orthogonality condition on R,

R = I = R R

. (60)

The condition above is indeed a condition of orthogonality, since we can observe from it

that, by writing R = [r

, r

] and substituting above, the column vectors r

of R, with

i ∈ {1, 2, 3}, are of unit length and orthogonal to each other,

, r

i = r

= 1

, r

i = r

= 0 , if i 6= j .

The set of transformations keeping vector norms and angles is for this reason called the

Orthogonal group, denoted O(3). The orthogonal group includes rotations (which are rigid

motions) and reﬂections (which are not rigid). The notion of group here means essentially

(and informally) that the product of two orthogonal matrices is always an orthogonal

matrix,

and that each orthogonal matrix admits an inverse. In eﬀect, the orthogonality

condition (60) implies that the inverse rotation is achieved with the transposed matrix,

−1

= R

. (61)

The eﬀect of the double cover needs to be considered when performing interpolation in the space of

rotations. This is however easy, as we will see in Section 2.7.

Let Q

and Q

be orthogonal, and build Q = Q

. Then Q

Q = Q

= Q

= I.

Adding the relative orientations condition (56) guarantees rigid body motion (hence

discarding reﬂections), and results in one additional constraint on R,

det(R) = 1 . (62)

Orthogonal matrices with positive unit determinant are commonly referred to as proper

or special. The set of such special orthogonal matrices is a subgroup of O(3) named the

Special Orthogonal group SO(3). Being a group, the product of two rotation matrices is

always a rotation matrix.

2.3.1 The exponential map

The exponential map (and the logarithmic map, which we see in the next section) is a

powerful mathematical tool for working in the rotational 3D space with ease and rigor. It

represents the entrance door to a corpus of inﬁnitesimal calculus suited for the rotational

space. The exponential map allows us to properly deﬁne derivatives, perturbations, and

velocities, and to manipulate them. It is therefore essential in estimation problems in the

space of rotations or orientations.

Rotations constitute rigid motions. This rigidity implies that it is possible to deﬁne

a continuous trajectory or path r(t) in SO(3) that continuously rotates the rigid body

from its initial orientation, r(0), to its current orientation, r(t). Being continuous, it is

legitimate to investigate the time-derivatives of such transformations. We do so by deriving

the properties (60) and (62) that we have just seen.

First of all, we notice that it is impossible to continuously escape the unit determinant

condition (62) while satisfying (60), because this would imply a jump of the determinant

from +1 to −1.

Therefore we only need to investigate the time-derivative of the orthogo-

nality condition (60). This reads

R) =

R + R

R = 0 , (63)

which results in

R = −(R

, (64)

meaning that the matrix R

R is skew-symmetric (i.e., it is equal to the negative of its

transpose). The set of skew-symmetric 3 × 3 matrices is denoted so(3), and receives the

name of the Lie algebra of SO(3). Skew-symmetric 3 × 3 matrices have the form,

[ω]





0 −ω

−ω





; (65)

Notice that reﬂections satisfy |R| = det(R) = −1, and do not form a group since |R

| = 1 6= −1.

See footnote 6 for O(3) and add this for SO(3): let |R

| = |R

| = 1, then |R

| = |R

||R

| = 1.

Put otherwise: a rotation cannot become a reﬂection through a continuous transformation.

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误差状态卡尔曼滤波器中的四元数动力学

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