使用单位四元数求解绝对定向的封闭形式解

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"Closed-Form Solution of Absolute Orientation Using Unit Quaternions.pdf" 这篇论文详细探讨了使用单位四元数解决绝对定向问题的闭合形式解法。绝对定向在计算机视觉、机器人学以及SLAM（Simultaneous Localization and Mapping，同时定位与建图）领域中具有重要意义。SLAM是自动驾驶、无人机导航和增强现实等技术的核心，它涉及到如何在一个未知环境中实时构建地图并确定自身位置。文章作者Berthold K. P. Horn是麻省理工学院的知名学者，他在该领域的研究成果丰富，引用次数高达27,547次。他的研究项目包括交通流稳定性控制，这可能与使用传感器数据进行动态系统定位有关，与SLAM技术有密切联系。单位四元数是一种数学工具，常用于表示三维空间中的旋转。相比于欧拉角或旋转矩阵，四元数避免了万向节死锁问题，并且具有更高效的运算特性。在绝对定向问题中，我们需要找到两个坐标系之间的相对旋转，以便将一个坐标系中的物体位置转换到另一个坐标系中。四元数的闭合形式解可以简洁、稳定地完成这个任务。论文中提到的方法可能包括以下步骤： 1. **数据获取**：通过传感器（如IMU、激光雷达或相机）收集两组坐标系之间的对应点。 2. **预处理**：校准传感器数据，消除噪声和误差。 3. **最小化误差函数**：构建一个描述对应点之间差异的误差函数，例如平移和旋转的余弦距离。 4. **四元数解法**：利用四元数的代数性质，寻找最小化误差函数的旋转四元数，这通常涉及线性代数中的最小二乘法。 5. **应用旋转**：使用求得的四元数对点集进行旋转，实现坐标系间的转换。这种方法的优势在于，它提供了精确且无歧义的旋转表示，并且可以通过解析解快速计算，适合实时系统。在SLAM中，这种技术对于不断更新地图和机器人位置至关重要。 "Closed-Form Solution of Absolute Orientation Using Unit Quaternions"这篇论文详细阐述了如何利用单位四元数来高效解决三维空间中的绝对定向问题，这对现代自动化系统的定位和导航有着深远的影响。

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630

Opt.

Soc.

Am.

A/Vol.

No.

4/April

1987

Y_. /

)1,- /

, \Zr,

A, I

,/ oD

_ \ \

\ -

Fig. 1. The

coordinates

of a number

of points

is measured

in two

different

coordinate

systems.

The

transformation

between

the

two

systems

is to

be found.

expect

to be

able to find

a transformation

that maps

the

measured

coordinates

of points

in one system

exactly

into

the measured

coordinates

these points

in the

other.

In-

stead, we

minimize

the sum

of squares

of residual

errors.

Finding

the

best set

of transformation

parameters

is not

easy.

In practice,

various

empirical,

graphical,

and numeri-

cal procedures

are in use.

These

are iterative

in nature.

That is,

given an

approximate

solution,

such a method

leads

to a better,

but still

imperfect,

answer.

The iterative

meth-

is applied

repeatedly

until the

remaining

error is

negligi-

ble.

At times,

information

is available

that

permits

one

obtain so

good an

initial guess

of the transformation

parame-

ters that

a single step

of the

iteration brings

one

close enough

to the true

solution

of the least-squares

problem to

eliminate

the need

for further

iteration

in a practical

situation.

Closed-Form

Solution

In this

paper I

present

a closed-form

solution

the least-

squares

problem

of absolute

orientation,

one that

does not

require iteration.

One advantage

of a

closed-form

solution

that it provides

one

in a single

step with

the best

possible

transformation,

given the

measurements

the points

in the

two coordinate

systems.

Another

advantage

is that

one

need

not find

a good

initial

guess, as

one does

when

iterative

method

is used.

give the solution

a form in

which unit

quaternions

are

used

to represent

rotations.

The

solution

for the desired

quaternion

is shown

to be the

eigenvector

of a symmetric

4 matrix

associated

with

the most

positive

eigenvalue.

The

elements

of this

matrix

are simple

combinations

of sums

products

of corresponding

coordinates

of the

points.

find

the eigenvalues,

a quartic

equation

has to be

solved

whose

coefficients

are

sums of products

elements

of the

matrix.

It is

shown that

this quartic

is particularly

simple,

since

one of its

coefficients

is zero.

It simplifies

even

when one

or the other

of the

sets of measurements

is copla-

nar.

E. Orthonormal

Matrices

While

unit quaternions

constitute

an elegant

representation

for

rotation, most

of us

are more

familiar with

orthonormal

matrices

with positive

determinant.

Fortunately,

the

ap-

propriate

3 X 3

rotation matrix

can

be easily

constructed

from the four

components

of the

unit quaternion,

as is shown

Subsection

3.E. Working

directly

with

matrices

is diffi-

cult

because

of the need

to deal

with six

nonlinear

con-

straints

that ensure

that

the matrix

is orthonormal.

closed-form

solution

for the

rotation

matrix

using

manipula-

tions

matrices

will

be presented

in a subsequent

paper.

This

closed-form

solution

requires

the computation

the

square

root

of a symmetric

3 X 3

matrix.

F. Symmetry

of the

Solution

Let us

call the

two coordinate

systems

"left"

and "right."

desirable

property

of a

solution

method

is that,

when

applied

the problem

finding

the best

transformation

from

the

right to

the left

system,

gives the

exact

inverse

of the

best

transformation

from

the

left to the

right

system.

I show

Subsection

2.D

that the

scale

factor

has to

be treated

particular

way

to guarantee

that this

will

happen.

Symme-

try

is guaranteed

when

one uses

unit

quaternions

repre-

sent

rotation.

SOLUTION

METHODS

As we

shall

see, the

translation

and the

scale

factor

are easy

to determine

once

the rotation

known.

The

difficult

part

the problem

finding

the rotation.

Given

three

noncol-

linear

points,

we can

easily

construct

a useful

triad

each of

the

left

and the

right

coordinate

systems

(Fig.

2).

Let the

origin

at the

first

point.

Take

the line

from

the first

the second

point

be the

direction

the new

x axis.

Place

the new

y axis

at right

angles

to the

new

x axis

in the plane

formed

by the

three

points.

The

new z

axis is

then made

be orthogonal

to the x

and y

axes, with

orientation

chosen

satisfy

the

right-hand

rule.

This

construction

is carried

out

both

left and

right

systems.

The

rotation

that

takes

one

of these

constructed

triads

into

the other

is also

the

rotation

that

relates

the two

underlying

Cartesian

coordinate

sys-

tems.

This

rotation

is easy

to find,

as we

show

below.

A. Selective

Discarding

Constraints

Let

the

coordinates

of the

three

points

in each

of the

two

coordinate

systems

be r

j, r

rl,

and

rr,,, rr,2,

rr,3,

respec-

tively.

Construct

= r

- rl,-

Then

= xi/

xllf

a unit

vector

in the

direction

of the

new

x axis

in the

left-

hand system.

Now let

= (rl,

rj,j) - [(rl,3

- r

,) . X

Fig. 2.

Three points

define

a triad.

Such

a triad can

be construct-

ed by

using

the left

measurements.

second

triad is

then con-

structed

from

the right

measurements.

The

required

coordinate

transformation

can be estimated

by finding

the

transformation

that

maps one

triad

into

the other.

This

method

does

not

use the

information

about

each

of the three

points

equally.

Berthold

K. P. Horn

,P%,

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使用单位四元数求解绝对定向的封闭形式解

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