手眼标定AX=XB论文注解版 Solving AX = XB on the Euclidean Group

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1.原文名称：Robot Sensor Calibration: Solving AX = XB on the Euclidean Group 2.原文下载自行搜 3.文件中补充了一些原文给出没有完全推导过程的推理思路，楼主不是学数学的所以不能保证一定是严谨的，仅供参考，便于理解。 4.论文中出现了很多关于李群、李映射的知识，建议可以看看《AMathematicalIntroductiontoRoboticManipulation-book》 该书中有严谨的数学推导 《手眼标定AX=XB论文注解版》是一篇探讨机器人传感器校准问题的学术文章，主要关注在欧几里得群上的方程AX=XB的求解。该问题在机器人技术中至关重要，因为它涉及到腕部安装的传感器（如摄像头）相对于机器人手腕框架的精确位置和姿态的确定。通常，矩阵A描述了手腕框架经过任意运动后的相对位置和姿态，而B则表示传感器框架在相同运动后的相对位置和姿态。X则是传感器框架相对于手腕框架的位置和姿态。 文章提到了Frank C. Park和Bryan J. Martin的研究，他们利用李群理论的方法，得到了一个几何上可直观理解的封闭形式的精确解，以及在存在噪声时的封闭形式最小二乘解。这种解决方法不仅具有几何上的吸引力，而且由于欧几里得群存在一组规范坐标，使得问题的表述变得简单明了。 在机器人传感器校准中，需要通过多次机器人手臂的任意运动，收集数据并解决AX=XB来确定传感器的精确位置。早期的研究，如Shiu和Ahmad的工作，为这个方程提供了一个封闭形式的解以及唯一性条件。而Chou和Kame1则提出使用四元数来解决此问题。然而，本文作者通过李群理论提供了新的视角和解决方案。 李群理论是数学中处理连续对称性的有力工具，特别适用于描述旋转和平移这类连续变换。在这个问题中，李群理论帮助作者将欧几里得群中的矩阵运算转换成更直观的几何操作，从而简化了求解过程。 论文中提到的"李群"是一个具有群结构的光滑流形，它允许群元素之间的连续变换，而在"李映射"中，这些连续变换被定义为群到自身的光滑函数。在解决AX=XB时，李群理论的应用可以将矩阵运算转化为更直观的几何变换，使得即使非数学背景的读者也能理解解法的含义。 这篇论文对于理解机器人传感器校准中的核心问题提供了深入的数学解析，并为实际应用提供了一种有效且直观的解决方案。通过引入李群理论，作者不仅解决了实际工程问题，也为相关领域的研究者提供了理论基础和新的思考方向。对于想要深入了解机器人传感器标定或对李群理论感兴趣的读者来说，这是一篇不可多得的资源。
IEEE
TRANSACTIONS
ON
ROBOTICS
AND
AUTOMATION,
VOL.
10,
NO.
5,
OCTOBER
1994
717
Robot Sensor Calibration: Solving
AX
=
XB
on the Euclidean Group
Frank C. Park and Bryan
J.
Martin
Abstract-The equation
.-IS
=
SB
on the Euclidean group arises in
the problem of calibrating wrist-mounted robotic sensors. In this article
we derive, using methods of Lie theory, a closed-form exact solution that
can be visualized geometrically, and a closed-form least squares solution
when
.-I
and
B
are measured in the presence of noise.
I. INTRODUCTION
The equation
AS
=
SB
on the Euclidean group, where
A
and
Bare known and
S
is unknown, is of fundamental importance in the
problem of calibrating wrist-mounted robotic sensors. Typically the
matrix
-4
describes the position and orientation of the wrist frame
relative to itself after some arbitrary movement, and
B
describes the
position and orientation of the sensor (e.g., camera) frame relative to
itself after the same movement.
X
then describes the position and
orientation of the sensor frame relative to the wrist frame. Calibration
involves performing several arbitrary movements
of
the robot
arm
and
solving
A-Y
=
SB
for
-Y
to determine the precise location
of
the
sensor. While solutions to this equation have been studied when
A
and
B
are general
n
x
ii
matrices (see, e.g., Gantmacher
[SI),
in
robotic applications one is only interested in solutions that belong to
the
Euclidean
group.
Shiu and Ahmad [8] first motivate this equation in the context
of robot sensor calibration, and provide a closed-form solution and
conditions for its uniqueness. Chou and Kame1 [3] present a method
for solving this equation using quatemions. In this paper we present
both exact and least-squares solutions to this equation using methods
of Lie group theory. The principal advantage of this approach, aside
from its geometric appeal, is that there exists a set of
canonical
coordinates
for the Euclidean group that leads to
a
particularly simple
characterization of the solutions to
AX
=
XB.
The solution can be
expressed explicitly and also admits a simple geometric visualization.
Because
-4-1-
=
-XB
has a one-parameter family of solutions
(as first shown by Shiu and Ahmad
[8]),
two pairs of
(Az,
B,)
satisfying certain constraints are required in order to obtain a unique
solution. Unfortunately, for sensor calibration applications some
noise
is usually present in the measured values of
A
and
B,
so
that conditions for existence of a solution may not be satis-
fied. A more practical approach is to make several measurements
{
(-41.
B1).
(-42.
BZ
).
.
. .
.
(-AL.
BI
)},
and to find an
.X
that mini-
mizes the error criterion
where
d(
..
.)
is some distance metric on the Euclidean group. Using
the canonical coordinates for Lie groups the above minimization
problem can be recast into a least-squares fitting problem that
admits a simple and explicit solution. Specifically, given vectors
XI.
23..
.
.
.
SI.
and
yl.
y2.
. .
.
,
yr.
in Euclidean n-space, NBdas
[6]
provides explicit expressions for the orthogonal matrix
0
and trans-
Manuscript received October 21, 1992; revised September 24, 1993.
The
authors are with the Department of Mechanical and Aerospace Engi-
IEEE
Log
Number 9403328.
neering, University of Califomia at Irvine, Irvine, CA 92717 USA.
lation
b
that minimize
k
17
=
(I@S<
+
b
-
~21)~
1=1
The best values of
8
and
b
turn out to depend only on the matrix
M
=
.rz
y:.
By applying the canonical coordinates and this result
a “best-fit’’ solution to
A+Y
=
XB
can be obtained.
The paper is organized as follows. In Section
I1
we examine the Lie
group structure of the Euclidean group, and derive explicit formulas
for the canonical coordinates. In Section
111
we derive closed-form
exact solutions to the equation
AX
=
XB,
and in Section IV we
present a least-squares solution given a set of noisy measurements
for
A
and
B.
11. THE EUCLIDEAN
GROUP
For our purposes it is sufficient to think
of
SE(3), the Euclidean
group of rigid-body motions, as consisting of matrices of the form
[::
‘;I
where
0
E
SO(3) and
b
E
R3.
Here SO(3) denotes the group of
3
X
3
rotation matrices. SE(3) has the structure of both a differentiable
manifold and an algebraic group, and is an example of a
Lie group.
Some well known examples of matrix Lie groups include Gl(n),
the general linear group of
n
x
n
nonsingular matrices, and Sl(n),
the special linear group of
n
x
n
nonsingular matrices with unit
determinant.
Associated with every Lie group is its
Lie algebra.
In general
a Lie algebra is a vector space,
V,
together with a bilinear map
[
,
]
:
V
x
1,’
+
L7
(called the Lie
bracket)
that satisfies, for every
7,
p,
U
E
(i)
[v,
171
=
0,
and (ii)
[[v,
PI,
~1+[[4
71,
@.]+[b.
VI,
71
=
0.
The Lie algebra of S0(3), denoted so(3), consists of the
3
X
3
skew-symmetric matrices of the form
0
-.d3
LJ2
[
LJ3
-;l]
5
[-.I
-
&‘2
Observe from this definition that so(3) can be identified with
R3.
The
Lie algebra of SE(3), denoted se(3), consists of the
4
X
4
matrices
of the form
b1
4
where
[w]
E
so(3) and
7’
E
R3.
For both so(3) and se(3) (and for
general matrix Lie algebras) the Lie bracket is given by the matrix
commutator:
[A,
B]
=
AB
-
BA.
A fundamental concept related to Lie groups is the
exponential
mapping.
Given a matrix Lie group
G
and its corresponding matrix
Lie algebra
g,
the exponential mapping is the map
exp
:
g
+
G
defined by the matrix exponential:
exp
;I
=
I
+
A
+
&A2
+
. . .
for
A
E
g.
Over some open set
U
C
g
containing
0
the mapping
exp
:
U
+
G
is a diffeomorphism.’ The exponential therefore
defines local coordinates over some neighborhood of the identity in
G;
Chevalley [2] calls these coordinates the
canonical coordinates
(with respect to a particular basis).
In the remainder of this section we derive explicit formulas for
the canonical coordinates on SE(3) and its subgroup SO(3). One of
‘A
difSeeomorphism
is a differentiable 1-1 and onto mapping whose inverse
is also differentiable.
1042-296X/94$04.00
0
1994 IEEE

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