机器人姿态分析：四元数、误差与数据融合详解

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在"机器人运动姿态分析基本知识"这一主题中，文章涵盖了多个关键知识点，对于理解和设计机器人运动控制系统的工程师和技术人员具有重要价值。以下是主要内容的详细解析： 1. **四元数基本知识**：四元数是一种数学工具，用于描述三维空间中的旋转。它扩展了复数的概念，提供了一种高效且无奇异性的方法来表示和计算旋转。文章首先介绍了四元数的定义，包括它的代数特性，如单位四元数表示纯旋转，以及它们如何与旋转矩阵和欧拉角进行转换。 2. **旋转基本概念**：这部分涵盖了旋转的基本概念，包括旋转轴、旋转角度以及旋转变换。理解这些概念有助于在机器人运动规划中正确地组合和执行关节运动，确保机械臂或无人机等设备按照预期移动。 3. **运动姿态误差分析**：姿态误差分析是机器人控制中的核心部分，它涉及到实际运动与理想运动之间的偏差。通过研究姿态误差，工程师可以评估传感器的精度，设计有效的校准和补偿机制，以提高系统性能。 4. **数据融合算法 - 补充滤波基础**：文章提到了数据融合，特别是互补滤波技术，这是一种常用的方法，将来自不同传感器（如惯性测量单元，IMU）的数据集成，减少噪声并提高位置和姿态估计的准确性。互补滤波通过结合加速度计和陀螺仪的数据，减小了单个传感器的不一致性。 5. **Runge-Kutta方法**：这是一个数值积分方法，常用于数值模拟和控制系统的设计，特别是在处理连续时间系统的动态模型时。Runga-Kutta方法用于估计系统状态随时间的变化，这对于实时更新机器人姿态预测和控制器的计算至关重要。在本文的后续部分，作者详细讨论了误差状态滤波器（Error-State Kalman Filter, ESKF）在惯性导航系统驱动的机器人系统中的应用，包括如何处理连续时间和离散时间下的系统动力学，以及如何融合互补感官数据，如全球角误差的处理。此外，还介绍了几种数值积分方法，如欧拉法和改进的数值技术，以提高姿态估计的精度。本文深入探讨了机器人运动姿态分析的核心要素，不仅提供了理论背景，还涵盖了实用技巧和常见问题的解决方案，对从事机器人控制和自主导航技术的研究者和开发者具有很高的参考价值。

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and are therefore proper rotation matrices in the R

space. To be more speciﬁc, they

represent isoclinic rotations. The fourth property, which was already seen in (22), states

that left- and right- isoclinic rotations commute.

Isoclinic rotations can be brieﬂy explained as follows. Contrary to what happens in R

(where vectors parallel to the axis of rotation do not move, and vectors perpendicular to

the axis rotate equally in a plane perpendicular to this axis), rotations in R

encompass two

independent rotations in two orthogonal 2-planes of the 4-space. This means that every

4-vector of each of these 2-planes rotates in its own plane, and is not aﬀected by rotations

in the other plane. These 2-planes are for this reason called ‘invariant’. Isoclinic rotations

are those where the angles of rotation in the two invariant planes have the same magnitude.

Then, when the two angles have the same sign, we speak of left-isoclinic rotations; when

they have opposite signs, we speak of right-isoclinic rotations.

Given a unit quaternion q = e

uθ/2

, representing a rotation in R

of θ around u, the

matrix [q]

is a left-isolinic rotation in R

corresponding to the left-multiplication by the

quaternion q, and [q]

is a right-isolinic rotation, corresponding to the right-multiplication

by the quaternion q. The angles of these isoclinic rotations are exactly of magnitude θ/2.

Then, the expression (68) represents two chained isoclinic rotations, one left- and one

right-, to the 4-vector (0, x)

, each by half the desired rotation angle in R

. In one of the

invariant 2-planes of R

, the two half angles cancel out, because they have opposite signs.

In the other plane, they sum up to yield the total rotation angle θ. If we deﬁne from (68)

the resulting rotation matrix, R

, one easily realizes that (see also (73)),

, [q

∗

]

[q]

= [q]

∗

]



1 0

0 R



, (74)

where R is the rotation matrix in R

, which clearly rotates the R

subspace of R

, leaving

the fourth dimension unchanged.

This discourse is far beyond the scope of the present document. We include it here just

as a means for providing yet another way to interpret rotations by quaternions, with the

hope that the reader grasps more intuition about its mechanisms. The interested reader is

suggested to consult the appropriate literature on isoclinic rotations in R

1.5 Quaternion conventions. My choice.

There are several ways to determine the quaternion. They are basically related to four

binary choices:

• The order of its elements — real part ﬁrst or last:

q =





vs. q =





. (75)

• The multiplication formula — deﬁnition of the quaternion algebra:

ij = −ji = k vs. ji = −ij = k , (76a)

which correspond to diﬀerent handedness, respectively:

right-handed vs. left-handed , (76b)

which means that, given a rotation axis u, one quaternion q{uθ} rotates vectors an

angle θ around u using the right hand rule, while the other quaternion uses the left

hand rule.

• The function of the rotation operator — rotating frames or rotating vectors:

Passive vs. Active. (77)

• In the passive case, the direction of the operation — local-to-global or global-to-local:

global

= q ⊗ x

local

⊗ q

∗

vs. x

local

= q ⊗ x

global

⊗ q

∗

(78)

This variety of choices leads to 12 diﬀerent combinations. Historical developments have

favored some conventions over others (Chou, 1992; Yazell, 2009). Today, in the available

literature, we ﬁnd many quaternion ﬂavors such as the Hamilton, the STS

, the JPL

, the

ISS

, the ESA

, the Engineering, the Robotics, and possibly a lot more denominations.

Many of these forms might be identical, others not, but this fact is rarely explicitly stated,

and many works simply lack a suﬃcient description of their quaternion with regard to the

four choices above.

The two most commonly used conventions, which are also the best documented, are

Hamilton (the options on the left in (75–78)) and JPL (the options on the right, with the

exception of (77)). Table 1 shows a summary of their characteristics. JPL is mostly used

in the aerospace domain, while Hamilton is more common to other engineering areas such

as robotics —though this should not be taken as a rule.

My choice, which has been taken as early as in equation (2), is to take the Hamilton

convention, which is right-handed and coincides with many software libraries of widespread

use in robotics, such as Eigen, ROS, Google Ceres, and with a vast amount of literature

on Kalman ﬁltering for attitude estimation using IMUs (Chou, 1992; Kuipers, 1999; Pini´es

et al., 2007; Roussillon et al., 2011; Martinelli, 2012, and many others).

The JPL convention is possibly less commonly used, at least in the robotics ﬁeld. It

is extensively described in (Trawny and Roumeliotis, 2005), a reference work that has an

aim and scope very close to the present one, but that concentrates exclusively in the JPL

convention. The JPL quaternion is used in the JPL literature (obviously) and in key papers

by Li, Mourikis, Roumeliotis, and colleagues (see e.g. (Li and Mourikis, 2012; Li et al.,

2014)), which draw from Trawny and Roumeliotis’ document. These works are a primary

Space Transportation System, commonly known as NASA’s Space Shuttle.

Jet Propulsion Laboratory.

International Space Station.

European Space Agency.

Table 1: Hamilton vs. JPL quaternion conventions with respect to the 4 binary choices

Quaternion type Hamilton JPL

1 Components order (q

, q

) (q

, q

)

Algebra ij = k ij = −k

Handedness Right-handed Left-handed

3 Function Passive Passive

Right-to-left products mean

Local-to-Global Global-to-Local

Default notation, q q , q

q , q

Default operation x

= q ⊗ x

⊗ q

∗

= q ⊗ x

⊗ q

∗

source of inspiration when dealing with visual-inertial odometry and SLAM —which is

what we do.

In the rest of this section we analyze these two quaternion conventions with a little

more depth.

1.5.1 Order of the quaternion components

Though not the most fundamental, the most salient diﬀerence between Hamilton and JPL

quaternions is in the order of the components, with the scalar part being either in ﬁrst

(Hamilton) or last (JPL) position. The implications of such change are quite obvious and

should not represent a great challenge of interpretation. In fact, some works with the

quaternion’s real component at the end (e.g., the C++ library Eigen) are still considered

as using the Hamilton notation, as long as the other three aspects are maintained.

We have used the subscripts (w, x, y, z) for the quaternion components for increased

clarity, instead of the other commonly used (0, 1, 2, 3). When changing the order, q

will

always denote the real part, while it is not clear whether q

would also do —in some

occasions, one might ﬁnd things such as q = (q

, q

), with q

real and last, but in

the general case of q = (q

, q

), the real part at the end would be q

When passing

from one convention to the other, we must be careful of formulas involving full 4 × 4 or

3×4 quaternion-related matrices, for their rows and/or columns need to be swapped. This

is not diﬃcult to do, but it might be diﬃcult to detect and therefore prone to error.

1.5.2 Speciﬁcation of the quaternion algebra

The Hamilton convention deﬁnes ij = k and therefore,

= j

= k

= ijk = −1 , ij = −ji = k , jk = −kj = i , ki = −ik = j , (79)

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