A Study of Quaternion in Terms of Indicial Notation
Min

Chan Hwang
(
黃敏昌
)
1
and
Lih

Jier Young (
楊立杰
)
2
1
Department of Automation Engineering
,
Ta
Hwa Insti
tute of Technology,
No. 1
,
Dahu
a Rd.,
Qionglin Shiang
, Hsinc
hu
County
,
307
, Taiwan
,
R.O.C.
Tel:
03

592

7
700

2675, Fax:
03

592

1047
Email:
aemch@thit.edu.tw
2
Department of Applied Mathematics, Chung

Hua University,
No. 707, Sec. 2, Wufu Rd., Hsinchu, 30012, Taiwan, R.O.C.
Tel:
03

518

6392, Fax:
03

537

3771
Email:
young@chu.edu.tw
Abstract
The convention
al approach using unabridged format to manipulate the algebraic properties of quaternion is
very cumbersome.
In order to manipulate the algebra more easily or more effectively, w
e
attempt to use
the
indicial notation
in the quaternion
. Although some author
s did use the indicial not
at
ion to deal with quaternion,
they merely applied it to the pure quaternion
,
which contains the vector part without the scalar part. In this paper,
the quaternion
of which we take into account is
in general
form, i.e.
including
b
oth of
the scalar
part
and vector
part of the quaternion. A concise survey on quaternion properties with proofs using indicial notation for some
known results is p
resented here. Additionally, three
examples are used to illustrate the application of the
qua
ternion in the rigid motion and the robotics
etc
.
Keywords:
indicial notation
,
quaternion, rigid motion, robotics
1.
Introduction
The quaternion [1]
[2]
,
which is a generalization
of a complex number was invented by Hamilton in
1843. He discovered t
hat the appropriate
generalization is one in which the scalar axis is left
unchanged whereas the vector axis is supplemented
by adding two further axes. The basic algebraic form
for a quaternion
q
is
0 1 2 3
ˆ
ˆ
ˆ
q q q i q j q k
.
(1)
In Einstei
n
’
s relativity theory [3], he introduced
indicial notations to simplify many calculations with
vectors.
The indicial notation has not only found its
vital success in physics but also in elasticity [4],
continuous mechani
cs, differential geometry, etc.
Since the quaternion contains the vector
components, it is natural to deal with the quaternion
algebra in terms of indicial notation. Although some
authors [2] did use the indicial notion to deal with
quaternion, they merely applied it to the pure
quaterni
on which contains the vector part without the
scalar part.
In this paper, the quaternion of which we
take into account is in general
form, i.e.
including
both of
the scalar part
and vector
part.
We
use the
to define the products
of the imaginary units
ˆ
ˆ
ˆ
,,
i j k
. The conventional approach using components
and imaginary units to manipulate the algebraic
properties of quaternion is cumbersome. In contrast
to the unabridged format, a concise survey on
quaternion properti
es with proofs using indicial
notation for some known results is presented here. As
you will see, the indicial notation makes the algebra
structure transparent and easy to be managed. Some
derivations will be shown in detail to help the
audience take a gli
mpse of the efficiency of this
approach as compared to the
unabridged
approach.
2
. A Brief of Indicial Notation
There are two categories of indices, i.e. the free
indices and the dummy indices. The free indices are
free to take any value while the dummy i
ndices are
summed over all
possible
values. In Einstein
summation convention, it is illegal to use the same
dummy index more than twice in a term. However,
we might encounter the cases of indices which repeat
themselves more than twice here. In order to a
void
any possible confusion, we would like to mak
e the
following distinctions.
2 2 2 2 2
1 2 3
3 3 3 3
1 2 3
2 2 2 2 2 2 2 2
1 2 3
4 4 4 4
1 2 3
( ) ( ) ( )
( ) ( ) ( )( )
( )
i i i j i
i
i i j
i
a a a a a a a a
a a a a
a a a a a a
a a a a
In other words, the dummy indices only prevail
in the local sense
, i.e.
one dummy variable appeared
in two different brackets treated as two individual
du
mmy variables. This convention could prevent the
prodigious growth in indicial notations.
The following
identity is extremely
important and extensively used here.
ijk lmk il jm im jl
(2)
where the Kronecker delta
and Levi

Civita symbol
are defined below.
0 if
1 if
ij
i j
i j
1 if 123,231,312
0 if any two indices are the same
1 if 321,213,132
ijk
ijk
ijk
3
. Definition of Quaternion Algebra
R
oughly speaking, algebra is a linear space
over
a field that admits a product operation. A precise
definition is stated below.
De
finition
1:
Let
S
be a finite

dimensional linear
space over a field
F
.
If
,,
a b c S
and
F
,
S
is called an algebra if it possesses the following
properties.
(i)
( ) ( ) ( )
ab a b a b
(ii)
( ), ( )
a b c ab ac b c a bc ca
We use the indicial notation to rewrite the
representation of
q
as
0
ˆ
i i
q q q e
(3)
where the subscript

i has its value over {1,2,3} and
follows the rule of Einstein summation.
Because
S
is a linear space, the imaginary uni
ts
ˆ
i
e s
sta
nd
for
the base vectors
,
ˆ
ˆ
ˆ
,,
i j k
.
To determine
the multiplication rules, we assign the operations on
the
ˆ
i
e
to be
ˆ
ˆ
ˆ
i j ij ijk k
e e e
.
(4)
The addition rule of quaternion
numbers is
defined in terms of indicial notation.
0 0
ˆ
( ) ( )
i i i
a b a b a b e
(5)
The product of two quaternion numbers can be
obtained by the rule for multiplying sums as follows.
0 0 0 0
ˆ
ˆ
ˆ
( )
i i i i j i j
ab a b a b b a e a b e e
(6)
Introducing (4) into the equation (6), we h
ave
0 0 0 0
ˆ
( )
i i k k ijk i j k
ab a b a b a b b a a b e
(7)
If the Equation (7) is
unabridged
, it is identical
to the result obtained by
the conventional approach,
i.e.
0 0 1 1 2 2 3 3
0 1 0 1 2 3 3 2
0 2 0 2 3 1 1 3
0 3 0 3 1 2 2 1
ˆ
( )
ˆ
( )
ˆ
( )
ab a b a b a b a b
b a a b a b a b i
b a a b a b a b j
b a a b a b a b k
.
(8)
The p
roperty (i), (ii) of definition
1 can be easily
verified
using
the equation (
7
)
with the argument of
distributive and communicative property of the real
numbers, i.e.
0 0 0
0 0 0
( ) ( ) ( )
ˆ
[ ( ) ( ) ( )]
i i i
k k k ijk i j j k
a b c a b c a b c
a b c b c a a b c e
0 0 0 0
0 0 0 0
ˆ
( )
ˆ
( )
i i k k ijk i j k
i i k k ijk i j k
a b ab a b b a ab e
a c a c a c c a c b e
ab bc
(9)
Likewise, the rest can be proved to show that the
q
uaternion indeed is an algebra.
4
. Associative Normed Algebra
Ano
ther
important property of quaternion is that
it is not only associative but also a division.
Moreover, the absolute value of a product is the
product of the absolute values of the
factors
.
Theorem
1:
Associative Normed Algebra
The quaternion constitutes a
n associative normed
algebra, i.e.
(i)
0 0 0 0 0 0
0 0 0 0 0 0
( ) ( )
( )
[
i i i i i i ilm i l m
k k k k i i i k i i i k
a bc ab c
a b c a bc ab c abc abc
a b c a b c a b c a bc ab c abc
0 0 0
ˆ
( ) ]
i j k i j i j i j k
a b c a b c a b c e
(10)
(ii)
2 2 2
0 0 0 0
( ) ( ) ( ) ( )( ).
k k i i j j
ab a b
a b a b b a a a b b
(11)
Proof
:
The part (i) is proved by expressing both sides
of the equality in terms of indicial notation and
showing tha
t they are identical to each other.
0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0
0 0 0 0 0 0
0
ˆ
( ) [ ( ) ]
( ) ( )
[( ) ( )
ˆ
( )]
[
i i k k ijk i j k
i i i i i lmi l m
i i k k k lmk l m
ijk i j j srj s r k
i i i i i i lmi i l m
k
a bc a b c bc b c c b bc e
a b c bc a b c c b bc
b c bc a a b c c b bc
a b c c b b c e
a b c a bc a b c a c b a bc
a b c
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
ˆ
( ) ]
( )
[
ˆ
( )]
k i i k k lmk l m
ijk i j i j ijk srj i s r k
i i i i i i ilm i l m
k k k k i i i k i i i k
ijk i j i j i j k
a bc a b c a c b a bc
a b c a c b a b c e
a b c a bc a b c a bc a bc
a b c a b c a b c a bc a b c a bc
a bc a b c a b c e
0 0 0 0
0 0 0 0 0
ˆ
( ) [ ( ) ]
( ) ( )
i i k k ijk i j k
i i i i lmi l m i
ab c a b a b a b b a a b e c
a b a b c a b b a a b c
0 0 0 0 0
0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
0 0
[( ) ( )
ˆ
( ) ]
[
ˆ
( ) ]
i i k k k ijk i j
ijk i i lmi l m j k
i i i i i i lmi l m i
k k k i i k ijk i j
ijk i i j ijk lmi l m j k
a b a b c c a b b a a b
a b b a a b c e
a b c a bc a b c a bc a b c
a b c a b c a b c a bc a b c
a b b a c a b c e
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
( )
[
ˆ
( )]
i i i i i i lmi l m i
k k k k j j j k j i i k
ijk i j i j i j k
a b c a bc a b c a bc a b c
a b c a b c a b c a b c a b c a bc
a bc a b c a b c e
The part (ii) is proved in a similar fashion.
Using (7) to obtain
2 2
,
a b
, we have
2
*
0 0 0 0
0 0
ˆ
( )
i i k k ijk i j k
i i
a a a a a a a a a a a a a e
a a a a
a
nd likewise
2
*
0 0
i i
b b b b b bb
.
Hence,
2 2
* *
2
0 0 0 0 0 0
( )( )
( ) ( )( ) ( )( ) ( )( )
i i i i i i i i
a b a a b b
a a a a bb a a b b a a bb
The right hand side of identity in (ii)
is
2
* 2
0 0
0 0 0 0
2 2
0 0 0 0
0 0
2 2
0 0 0 0
( ) ( ) ( )
( )( )
( ) ( )
2 ( )
( ) ( )
( )
i i
k k ijk i j k k lmk l m
i i k k
ijk i j k k ijk lmk i j l m
i i k k
il jm im jl i j l
ab ab ab a b a b
a b b a a b a b b a a b
a b a b a b b a
a b a b b a a b a b
a b a b a b b a
a b a
2 2
0 0 0 0
( ) ( )
m
i i k k i j i j i j j i
b
a b a b a b b a a b a b a b a b
As we develop the quadratic terms of the above
equation
, it is very easy to identify
that
2
( )
i i
a b
and
i i j j
aba b
cancel each other
. Hence,
2
2 2
0 0 0 0
2 2
0 0 0 0
2 2 2
0 0 0 0
( ) 2 ( )
( ) 2 ( )
( ) ( ) ( ) ( )( )
i i i i
k k k k i i j j i i j j
k k i i j j
ab a b a b a b a b
a b a b a b b a a a b b a ba b
a b a b b a a a b b
Q.E.D.
5. Non

Commutative Field
The quaternion resembles the real number in
many aspects except that it doesn
’
t possess order
structure and has no commutative property.
Therefore,
its quotients are defined as left quotient
and right quotient respectively.
Theorem
2: Quotient
(i)
The
left
quotient
of b by a
is defined as
ax b
where
2 2
0 0 0 0 0
/( )
ˆ
[ ( ) ]/( ) (12)
i i k k ijk i j k i
x ab aa
a b ab a b b a ab e a a
(ii) The
right
quotient
of b by a
is defined as
ya b
where
2 2
0 0 0 0 0
/( )
ˆ
[ ( ) ]/( ) (13)
i i k k ijk i j k i
y ba aa
a b ab a b b a ab e a a
The
condition
under which
the quaternion
is
commutative
is
c
o

linear on
the
vector part
of the
quaternions i.e.
ˆ
0
ijk i j k
a b e
.
Corollary
2

1: Inversion
Let
q
be a quaternion. Its inverse is equal to
2
1 * 2 2
0 0
ˆ
/( )/( )
i i i
q q q q qe q q
(14)
Note that
*
q
is the conjugate of the quaternion
q
and their vector
parts
are co

linear. Thus, there is no
distinction between left inverse and right inverse of a
quaternion.
6. Affine
/
H
omogeneous Transformation
The inner automorphism induced by a
quaternion has o
ne remarkable application
, i.e.
to
depict the rotation about a fixed axis.
Lemma
3

1: Automorphism
If
0
ˆ
i i
q q q e
is a unit quaternion, i.e.
1
q
, a pure
quaternion
ˆ
i i
x x e
through automorphism induced
by
q
is equal to the following.
1 * 2
0 0
ˆ
[( ) 2 2 ]
i i k j j k ijk i j k
qxq qxq q qq x q x q q q x e
(15)
Proof:
Since
q
is a unit quaternion, it is
obvious
that
1 *
q q
by corollary
2

1. In the sequel,
we apply
equation (
7
),
identity a
nd rearrange the
dummy indices.
*
0 0
ˆ
ˆ
[ ( ) ][ ]
j j k ijk i j k s s
qxq q x q x q x e q q e
0 0 0
0
2
0 0 0
0
2
0 0
ˆ
ˆ
( )
ˆ
ˆ
( )
ˆ
( )
ˆ
( )( )
(
j j k ijk i j k j j s s
s k ijk i j k s
j j k ijk i j j j k k
s k s ijk i j ks ksr r
k ijk i j
q x q q q x q x e q x q e
q q x q x e e
q x q q x q q x q x q e
q q x q q x e
q x q q x
0
2
0 0 0
2
0 0
ˆ
)
ˆ
( )
ˆ
( )
ˆ
( )
ˆ
( 2 )
(
j j k k
ksr s k srk ijk s i j r
k ijk i j j j k rsk s r k
si rj sj ri s i j r
k j j k ijk i j k
i i r j r j
q x q e
q q x q q x e
q x q q x q x q q q x e
q q x e
q x q x q q q x e
q q x q q x
2
0 0
ˆ
)
ˆ
[( ) 2 2 )]
r
i i k j j k ijk i j k
e
q q q x q x q q q x e
Q.E.D.
Supposed that a
pure quaternion
x
rotates about
a
pure and unit q
uaternion
p with a angle
, its new
position can be obtained by means of
the
automorphism
as follows.
1
y qxq
(16)
where
cos sin
2 2
q p
and
ˆ
i i
p me
fo
r
1
i i
mm
.
Theorem
3
:
Affine Transformation
Supposed that
q
is a unit quaternion and
b
is a pure
quaternion, the affine transformation of a pure
quaternion
x
induced by
q
and
b
can be defined as
follows.
*
2
0 1 1 2 1 0 3 3 1 0 2
2
2 1 0 3 0 2 2 3 2 0 1
2
3 1 0 2 3 2 0 1 0
(17)
2 2 2 2 2
2 2 2 2 2
2 2 2 2
i i
i i
i i
y qxq b
q q q q q q q q q q q q q
q q q q q q q q q q q q q
q q q q q q q q q q q
1 1
2 2
3 3 3 3
2
x b
x b
q q x b
The quaternion gives a concise representation of
the rigid motion. A counterpart of quaternion
representation is the homogeneous transformation [5]
[6] which is extensively used in robotic systems. The
following coro
llary, a conse
quence of theorem 3
,
states the conversion from a
n
affine transformation to
a homogeneous transform
ation
.
Corollar
y
3

1: Homogeneous Transformation
Given specific quaternion
s,
1 2 3
0
T
b b b b
and
cos sin sin sin
2 2 2 2
T
q l m n
,
the affine
transform
ation shown in theorem
3 can be
represented by the corresponding
homogeneous
transformation
y Tx
(18)
where
2 2 2
1
l m n
1 2 3
1 2 3
1
1
T
T
y y y y
x x x x
2
1
2
2
2
3
(1 cos ) cos (1 cos ) sin (1 os ) sin
(1 cos ) sin (1 cos ) cos (1 os ) sin
(1 os ) sin (1 os ) sin (1 cos ) cos
0 0 0 1
l lm n ln c m b
lm n m mn c l b
T
ln c m mn c l n b
Proof:
Plu
nge in each component of q and b to the
equation (17) and simplify them with the
trigonometric functions, i.e.
2 2 2 2 2 2 2 2
0 1 1
2
2 cos sin ( ) 2 sin
2 2 2
cos (1 cos ).
i i
q q q q q l m n l
l
Q.E.D.
As a result of the corollary
3

1,
one translational
transformation
and three rotational transf
ormations
with respect to
x
,
y
,
z
can be obtained
as
1 0 0
0 1 0
(,,)
0 0 1
0 0 0 1
a
b
Trans a b c
c
,
1 0 0 0
0 cos sin 0
(,)
0 sin cos 0
0 0 0 1
Rot x
,
cos 0 sin 0
0 1 0 0
(,)
sin 0 cos 0
0 0 0 1
Rot y
,
cos sin 0 0
sin cos 0 0
(,)
0 0 1 0
0 0 0 1
Rot z
.
7. Applications
In kinematics, the treatment of every problem is
generally req
uired to describe a motion with respect
to the inertial frame.
For instance, c
onsider a
broom car system as shown below. A rigid rod of
length L pivoted at the top of the car can swing as the
car moving horizontal
ly
.
Figure
1
A Broom Car System
In the a
spect of homogeneous transformation,
we need to define three reference frames attached to
the system. The point at the
tip of the rode is denoted
as
3
P
and
0
P
to indicate its position with respect
to frame

3 and inertial frame, respectively. Three
homogeneous transformation matrices are defined as
2
3
(,)
T Rot z
,
1
2
(0,,0)
T Trans h
,
0
1
(,0,0)
T Trans x
.
Then, we have
0 0 1 2 3
1 2 3
sin
cos
0
1
L x
L h
P T T T P
where
3
0 0 1
T
P L
.
I
nstead of three transformation matrices
,
we
only need to define two quaternions
a
s the quaternion
is applied
, i.e.
q
for the axis of rot
ation and
b
for the
translation
.
3
2
ˆ
P Le
,
3
ˆ
cos sin
2 2
q e
,
1 2
ˆ
ˆ
b xe he
The following result identical to the previous one is
obtained using the affine transformation.
0 3 *
1 2
ˆ
ˆ
( sin ) ( cos )
P qP q b L x e L h e
.
Figure
2
A
Five

Jointed Robot
A slightly complicated application is to formulate
the kinematics for a five

jointed robot as show
n
above
.
Five pairs of quaternions encoding the rotation
s
and translation
s
are defined
for
perform
ing
a
sequence of affine transformations
, i.e.
5
5 5
5 2 5 1 2
ˆ
ˆ
ˆ
cos sin, , 0
2 2
q e b fe ee P
,
3 3
4 4
4 1 4 2 3 1 3 2
ˆ
ˆ
ˆ
ˆ
cos sin, , cos sin,
2 2 2 2
q e b de q e b ce
,
2 2 1 1
2 3 2 2 1 3 1 3
ˆ
ˆ
ˆ
ˆ
cos sin, , cos sin,
2 2 2 2
q e b be q e b ae
,
where
a
,
b
,
c
,
d
,
e
,
f
are the known len
gth parameters
of the linkages
, and
0 5 * * *
1 2 3 4 5 5 5 4 4 3
* *
3 2 2 1 1
0 0 0
1 1 2 2 3 3
( ( ( ( ) )
) )
ˆ
ˆ
ˆ
P q q q q q P q b q b q
b q b q b
P e P e P e
,
where
0
1 1 1 2 1 2 3 4
1 2 3 4
1 2 3 1 2 3 1 2
sin cos( ) [sin( )
2
sin( )]
[sin( ) sin( )] sin( )
2
e
P b f
d
c
0
2 1 1 2 1 2 3 4
1 2 3 4
1 2 3 1 2 3 1 2
cos sin( ) [cos( )
2
cos( )]
[cos( ) cos( )] cos( )
2
e
P b f
d
c
0
3 3 4 3
sin( ) sin
P e d a
.
One
additional
example of solving a Lyapunov
like equation is illustrated below to
justify the
effectiveness of this
approach
.
The Lyapunov
equation often appears in the study of the
stability of
a control system.
*
0
a x xa q
(19)
where
,
a q
are known quaternions but
x
is a
unknown quaternion ready to be solved.
Due to the fact that the quaternion is not
a
commutative
field
, an equation as
shown
in (19)
can
not be solved
in a manner of
straightforward.
We begi
n to develop
the terms,
*
a x
and
xa
as follows.
*
0 0
0 0 0 0
ˆ
ˆ
( )( )
ˆ
( )
i i j j
i i k k ijk i j k
a x a a e x x e
a x a x a x a x a x e
(
20)
0 0
0 0 0 0
ˆ
ˆ
( )( )
ˆ
( )
j j i i
i i k k ijk i j k
xa x x e a a e
a x a x a x a x x a e
(21)
As a result of the indicial notion, the term
s in
(20),
(21) are
not only obtained in a way of great
efficiency but also
in
an
algebraic transpa
rency
so
that
we could easily render the following result.
*
0 0 0
ˆ
2 2( 2 )
k ijk i j k
a x xa a x a x x a e
(2
2
)
Hence,
*
0 0 0 0
0
ˆ
2 [2( ) ]
k ijk i j k k
a x xa q
a x q a x x a q e
(23)
By the
definition of the quaternion, a zero
quaternion implies that its scalar part and vector part
are all zero, i.e.
0 0 0
0
2 0
2 2 0
k ijk i j k
a x q
a x x a q
.
(24)
Thus
,
the solution of (19)
is obtained as follows.
0
0
0
2
q
x
a
2 2
1 0 1 1 1 2 0 3 2 1 3 0 2 3
2
0
1
{( ) ( ) ( ) }
2
x a a q a a a a q a a a a q
a a
2 2
2 1 2 0 3 1 0 2 2 2 3 0 1 3
2
0
1
{( ) ( ) ( ) }
2
x a a a a q a a q a a a a q
a a
2 2
3 1 3 0 2 1 2 3 0 1 2 0 3 3
2
0
1
{( ) ( ) ( ) }
2
x a a a a q a a a a q a a q
a a
8.
Conclusions
It is tedious to write long expressions wit
h lots
of components and imaginary units to manipulate the
quaternion. Thus, we introduce the indicial notation
and the Einstein summation to simplify the algebra
manipulation. In order to prevent the prodigious
growth in indicial notations, the dummy ind
ices
,
which
repeat themselves more than twice
,
are
permitted under the specification.
One practical application of the quaternion is to
depict the motion of a rigid body.
Its
counterpart
,
i.e.
homogeneous transformation
,
is extensively used
in robotic sy
stems.
A sequence of the affine
transformations in quaternion is equivalent to a
sequence of matrices
’
multiplication in the
homogeneous transformation.
The quaternion would
not only encode the rigid motion in a concise way
but also give the representation
in
close
agreement
with experience.
References
[1]
I.
L. Kantor
,
A.
S. Solodovnikov
,
"
Hypercomplex
Numbers,
"
Springer

Verlag
, 198
9
.
[2]
J. P. Ward
, “
Quaternions and Cayley Numbers
,”
Kluwer Academic Publishers, London, 1997
.
[3]
Albert Einstein,
“
The Mean
ing of Relativity,
”
Princeton University Press, Princeton, N.J.,
1956.
[4]
Arthur P. Boresi, Ken P. Chong
, "
Elasticity in
Engineering Mechanics,
"
Elsevier Science
Publishing Co., Inc.
, 19
87
.
[5]
Janez Funda, Russell H. Taylor
,
Richard P. Paul
"
On Homogeneo
us Transforms, Quaternions, and
Computational Efficiency,
"
IEEE Transactions on
Robotics and Automation,
Vol.
6
, No.
3
, pp.
382

388
,
June
199
0
.
[6]
Richard D. Klafter, Thomas A. Chmielewski,
Michael Negin,
"
Robotic Engineering An
Integrated Approach,
"
Prentice

Hall, Inc., New
Jersey
, 19
89
.
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