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 Institute of Technology,
No. 1, Dahua Rd., Qionglin Shiang, Hsinchu County, 307, Taiwan, R.O.C.
Tel: 03-592-7700-2675, 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 conventional 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, we attempt to use the
indicial notation in the quaternion. Although some authors did use the indicial notation 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 both 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 presented here. Additionally, three examples are used to illustrate the application of the
quaternion in the rigid motion and the robotics etc.
Keywords: indicial notation, quaternion, rigid motion, robotics
1. Introduction
Since the quaternion contains the vector
The quaternion [1] [2], which is a generalization
components, it is natural to deal with the quaternion
of a complex number was invented by Hamilton in
algebra in terms of indicial notation. Although some
1843.
appropriate
authors [2] did use the indicial notion to deal with
generalization is one in which the scalar axis is left
quaternion, they merely applied it to the pure
unchanged whereas the vector axis is supplemented
quaternion which contains the vector part without the
by adding two further axes. The basic algebraic form
scalar part. In this paper, the quaternion of which we
for a quaternion q is
take into account is in general form, i.e. including
He
discovered
that
the
q  q0  q1iˆ  q2 ˆj  q3kˆ .
(1)
both of the scalar part and vector part. We use the
In Einstein’s relativity theory [3], he introduced
   to define the products of the imaginary units
indicial notations to simplify many calculations with
iˆ, ˆj, kˆ . The conventional approach using components
vectors. The indicial notation has not only found its
and imaginary units to manipulate the algebraic
vital success in physics but also in elasticity [4],
properties of quaternion is cumbersome. In contrast
continuous mechanics, differential geometry, etc.
to the unabridged format, a concise survey on
3. Definition of Quaternion Algebra
quaternion properties with proofs using indicial
notation for some known results is presented here. As
Roughly speaking, algebra is a linear space over
you will see, the indicial notation makes the algebra
a field that admits a product operation. A precise
structure transparent and easy to be managed. Some
definition is stated below.
derivations will be shown in detail to help the
Definition 1: Let S be a finite-dimensional linear
audience take a glimpse of the efficiency of this
space over a field F.
approach as compared to the unabridged approach.
is called an algebra if it possesses the following
If a, b, c  S and   F , S
properties.
2. A Brief of Indicial Notation
(i)  (ab)  ( a )b  a ( b)
There are two categories of indices, i.e. the free
(ii) a(b  c)  ab  ac, (b  c)a  bc  ca
indices and the dummy indices. The free indices are
free to take any value while the dummy indices are
We use the indicial notation to rewrite the
representation of q as
summed over all possible values. In Einstein
q  q0  qi eˆi
(3)
summation convention, it is illegal to use the same
where the subscript-i has its value over {1,2,3} and
dummy index more than twice in a term. However,
follows the rule of Einstein summation.
Because S is a linear space, the imaginary units
eˆi s stand for the base vectors, iˆ, ˆj, kˆ . To determine
we might encounter the cases of indices which repeat
themselves more than twice here.
In order to avoid
any possible confusion, we would like to make the
the multiplication rules, we assign the operations on
the eˆi to be
following distinctions.
eˆi eˆ j   ij   ijk eˆk .
(ai2 )ai  (a12  a22  a32 )ai  (a 2j )ai
The addition rule of quaternion numbers is
ai3  a13  a23  a33
defined in terms of indicial notation.
(ai2 ) 2  ( a12  a22  a32 ) 2  (ai2 )( a 2j )
a  b  (a0  b0 )  (ai  bi )eˆi
(ai4 )  a14  a24  a34
obtained by the rule for multiplying sums as follows.
in the local sense, i.e. one dummy variable appeared
in two different brackets treated as two individual
ab  a0b0  (a0bi  b0 ai )eˆi  ai b j eˆi eˆ j
ab  a0b0  ai bi  (a0bk  b0 ak   ijk ai b j )eˆk (7)
prodigious growth in indicial notations.
If the Equation (7) is unabridged, it is identical
identity is extremely
important and extensively used here.
 ijk  lmk   il jm   im jl
(6)
Introducing (4) into the equation (6), we have
dummy variables. This convention could prevent the
(2)
where the Kronecker delta and Levi-Civita symbol
are defined below.
0 if i  j
1 if i  j
 ij  
 ijk
(5)
The product of two quaternion numbers can be
In other words, the dummy indices only prevail
The following   
(4)
 1 if ijk  123, 231,312

  0 if any two indices are the same
-1 if ijk  321, 213,132

to the result obtained by the conventional approach,
i.e.
ab  a0b0  a1b1  a2b2  a3b3
 (b0 a1  a0b1  a2b3  a3b2 )iˆ .
 (b0 a2  a0b2  a3b1  a1b3 ) ˆj
(8)
 (b0 a3  a0b3  a1b2  a2b1 )kˆ
The property (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.
(ab)c  [a0b0  ai bi  (a0bk  b0 ak   ijk ai b j )eˆk ]c
a(b  c)  a0 (b0  c0 )  ai (bi  ci )
[a0 (bk  ck )  (b0  c0 )ak   ijk ai (b j  c j )]eˆk
 a0b0  ai bi  (a0bk  b0 ak   ijk ai b j )eˆk
 (a0 b0  ai bi )c0  (a0bi  b0 ai   lmi al bm )ci
 [(a0b0  ai bi )ck  c0 (a0bk  b0 ak   ijk ai b j )
 a0 c0  ai ci  (a0 ck  c0 ak   ijk ci b j )eˆk (9)
  ijk (a0bi  b0 ai   lmi al bm )c j ]eˆk
 a0b0 c0  a0bi ci  ai b0 ci  ai bi c0   lmi al bm ci
 ab  bc
Likewise, the rest can be proved to show that the
quaternion indeed is an algebra.
 [ a0b0 ck  a0bk c0  ak b0c0  ai bi ck   ijk ai b j c0
  ijk (a0bi  b0 ai )c j   ijk  lmi al bm c j ]eˆk
 a0b0 c0  (a0bi ci  ai b0 ci  ai bi c0 )   lmi al bm ci
4. Associative Normed Algebra
[ak b0 c0  a0bk c0  a0b0ck  ak b j c j  a j bk c j  aibi ck
Another important property of quaternion is that
 ijk (a0bi c j  ai b0 c j  ai b j c0 )]eˆk
it is not only associative but also a division.
The part (ii) is proved in a similar fashion.
Moreover, the absolute value of a product is the
2
2
Using (7) to obtain a , b , we have
product of the absolute values of the factors.
Theorem 1: Associative Normed Algebra
a  a*a  a0 a0  ai ai  (a0 ak  a0 ak   ijk ai a j )eˆk
2
The quaternion constitutes an associative normed
 a0 a0  ai ai
algebra, i.e.
and likewise b  b*b  b0b0  bi bi .
2
(i)
a(bc)  (ab)c
 [ak b0c0  a0bk c0  a0b0ck  ak bi ci  ai bk ci  ai bi ck
 i j (k a0 b ci j a 0bi cj
a b  (a*a)(b*b)
2
 a0b0c0  (a0bi ci  ai b0ci  ai bi c0 )   ilm aibl cm
a i0b) ˆ]j c
ek
(10)
(ii)
Hence,
2
 (a0 a0 ) 2  (a0 a0 )(bi bi )  (ai ai )(b0b0 )  (ai ai )(bi bi )
The right hand side of identity in (ii) is
ab  (ab)* ( ab)  ( a0b0  ai bi ) 2
2
 ( a0bk  b0 ak   ijk ai b j )(a0bk  b0 ak   lmk al bm )
ab  a b
 (a0b0 )2  (a0bk ) 2  (b0 ak ) 2  (ai ai )(b j b j ).
(11)
 ( a0b0  ai bi ) 2  (a0bk  b0 ak ) 2
 2 ijk ai b j ( a0bk  b0 ak )   ijk  lmk ai b j al bm
 ( a0b0  ai bi ) 2  (a0bk  b0 ak ) 2
Proof:
The part (i) is proved by expressing both sides
of the equality in terms of indicial notation and
showing that they are identical to each other.
a (bc)  a[b0 c0  bi ci  (b0ck  c0bk   ijk bi c j )eˆk ]
 a0 (b0 c0  bi ci )  ai (b0ci  c0bi   lmi bl cm )
 [(b0 c0  bi ci )ak  a0 (b0ck  c0bk   lmk bl cm )
  ijk ai (b0 c j  c0b j   srj bs cr )]eˆk
 a0b0 c0  a0bi ci  aib0ci  ai c0bi   lmi aibl cm
 ( il jm   im jl ) ai b j al bm
 (a0b0  ai bi ) 2  (a0bk  b0 ak ) 2  ai b j ai b j  ai b j a j bi
As we develop the quadratic terms of the above
equation, it is very easy to identify that ( ai bi ) 2 and
ai bi a j b j cancel each other.
ab  (a0b0 ) 2  2a0b0 ai bi  (ai bi ) 2
2
(a0bk ) 2  2a0b0 ak bk  (b0 ak ) 2  ai ai b j b j  ai bi a j b j
 (a0b0 ) 2  (a0bk ) 2  (b0 ak ) 2  ( ai ai )(b j b j )
 [ak b0 c0  ak bi ci  a0b0ck  a0c0bk   lmk a0bl cm
  ijk ( ai b0 c j  ai c0b j )   ijk  srj ai bs cr ]eˆk
Q.E.D.
5. Non-Commutative Field
 a0b0 c0  (a0bi ci  aib0ci  aibi c0 )   ilm aibl cm
 [ak b0 c0  a0bk c0  a0b0ck  ak bi ci  aibk ci  aibi ck
  ijk ( a0bi c j  ai b0c j  ai b j c0 )]eˆk
Hence,
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
qxq*  [q j x j  (q0 xk   ijk qi x j )eˆk ][q0  qs eˆs ]
  q j x j q0  q0 (q0 xk   ijk qi x j )eˆk  q j x j qs eˆs
and right quotient respectively.
 qs (q0 xk   ijk qi x j )eˆk eˆs
Theorem 2: Quotient
  q j x j q0  (q02 xk  q0 ijk qi x j  q j x j qk )eˆk
(i) The left quotient of b by a is defined as
ax  b
 ( qs q0 xk  qs  ijk qi x j )( ks   ksr eˆr )
where
 (q02 xk  q0 ijk qi x j  q j x j qk )eˆk
x  ab /(aa)
 ( q0 ksr qs xk   srk  ijk qs qi x j )eˆr
 [a0b0  aibi  (a0bk  b0 ak   ijk aib j )eˆk ]/(a02  ai2 ) (12)
(ii)
 (q02 xk  q0 ijk qi x j  q j x j qk  q0 rsk qs xr )eˆk
 ( si rj   sj  ri )qs qi x j eˆr
The right quotient of b by a is defined as
ya  b
 (q02 xk  q j x j qk  2q0 ijk qi x j )eˆk
where
 ( qi qi xr  q j qr x j )eˆr
y  ba /(aa )
 [(q02  qi qi ) xk  2q j x j qk  2q0 ijk qi x j )]eˆk
 [a0b0  aibi  (a0bk  b0 ak   ijk aib j )eˆk ]/(a  a ) (13)
2
0
2
i
Q.E.D.
The condition under which the quaternion is
Supposed that a pure quaternion x rotates about
commutative is co-linear on the vector part of the
a pure and unit quaternion p with a angle  , its new
quaternions i.e.  ijk ai b j eˆk  0 .
position can be obtained by means of
Corollary 2-1: Inversion
the
automorphism as follows.
Let q be a quaternion. Its inverse is equal to
q 1  q* / q  (q0  qi eˆi ) /(q02  qi2 )
2
y  qxq 1
(14)

(16)

where q  cos 2  p sin 2 and p  mi eˆi for mi mi  1 .
Note that q* is the conjugate of the quaternion q
Theorem 3: Affine Transformation
and their vector parts are co-linear. Thus, there is no
Supposed that q is a unit quaternion and b is a pure
distinction between left inverse and right inverse of a
quaternion, the affine transformation of a pure
quaternion.
quaternion x induced by q and b can be defined as
6. Affine / Homogeneous Transformation
The
inner
automorphism
induced
by
a
quaternion has one remarkable application, i.e. to
depict the rotation about a fixed axis.
follows.
y  qxq*  b
(17)
 q  qi qi  2q1q1 2q2 q1  2q0 q3
2q3q1  2q0 q2   x1   b1 


2
  2q2 q1  2q0 q3 q0  qi qi  2q2 q2 2q3q2  2q0 q1   x2   b2 
 2q3q1  2q0 q2
2q3q2  2q0 q1 q02  qi qi  2q3q3   x3  b3 

2
0
Lemma 3-1: Automorphism
If q  q0  qi eˆi is a unit quaternion, i.e. q  1 , a pure
quaternion x  xi eˆi through automorphism induced
by q is equal to the following.
qxq1  qxq*  [(q02  qi qi ) xk  2q j x j qk  2q0ijk qi x j ]eˆk
(15)
Since q is a unit quaternion, it is obvious that
q 1  q * by corollary 2-1. In the sequel, we apply
dummy indices.
the rigid motion.
A counterpart of quaternion
representation is the homogeneous transformation [5]
[6] which is extensively used in robotic systems. The
following corollary, a consequence of theorem 3,
Proof:
equation (7),   
The quaternion gives a concise representation of
identity and rearrange the
states the conversion from an affine transformation to
a homogeneous transformation.
Corollary 3-1: Homogeneous Transformation
Given
specific
quaternions,
b   0 b1
b2
b3 
T
and q  cos  l sin  m sin  n sin  

2
2
transformation
,
the
affine
2
2
shown
T
in
generally required to describe a motion with respect
to the inertial frame.
theorem
3
can
For instance, consider a
be
broom car system as shown below. A rigid rod of
represented by the corresponding homogeneous
length L pivoted at the top of the car can swing as the
transformation
car moving horizontally.
y  Tx
(18)
 l 2 (1  cos  )  cos  lm(1  cos  )  n sin  ln(1  c os  )  m sin 

lm(1  cos  )  n sin  m2 (1  cos  )  cos  mn(1  c os  )  l sin 
T 
ln(1  c os  )  m sin  mn(1  c os  )  l sin  n 2 (1  cos  )  cos 

0
0
0

b1 

b2 
b3 

1
where l  m  n  1
2
y   y1 y2
2
2
y3 1
T
x   x1 x2 x3 1
T
Figure 1 A Broom Car System
In the aspect of homogeneous transformation,
we need to define three reference frames attached to
Proof:
Plunge in each component of q and b to the
the system. The point at the tip of the rode is denoted
the
as P 3 and P 0 to indicate its position with respect
trigonometric functions, i.e.



q02  qi qi  2q1q1  cos 2  sin 2 (l 2  m2  n 2 )  2l 2 sin 2
2
2
2
 cos   l 2 (1  cos  ).
Q.E.D.
to frame-3 and inertial frame, respectively. Three
equation
(17)
and
simplify
them
with
As a result of the corollary 3-1, one translational
transformation and three rotational transformations
with respect to x, y, z can be obtained as
1
0
Trans (a, b, c)  
0

0
0 0 a
1 0 b  ,
0 1 c

0 0 1
0
0
0
1
0 cos   sin  0 
,
Rot ( x,  )  
0 sin  cos  0 


0
0
1
0
 cos 
 0
Rot ( y,  )  
  sin 

 0
 cos 
 sin 
Rot ( z ,  )  
 0

 0
0 sin 
1
0
0 cos 
0
0
 sin 
cos 
0
0
0
0  ,
0

1
0 0
0 0  .
1 0

0 1
7. Applications
In kinematics, the treatment of every problem is
homogeneous transformation matrices are defined as
T32  Rot ( z,  ) , T21  Trans(0, h, 0) , T10  Trans( x, 0, 0) .
Then, we have
  L sin   x 
 L cos   h 

P 0  T10T21T32 P 3  


0


1


where P3   0 L 0 1
T
.
Instead of three transformation matrices, we
only need to define two quaternions as the quaternion
is applied, i.e. q for the axis of rotation and b for the
translation.
P 3  Leˆ2

,

q  cos  sin eˆ3 b  xeˆ1  heˆ2
2
2 ,
The following result identical to the previous one is
obtained using the affine transformation.
P 0  qP3q*  b  ( L sin   x)eˆ1  ( L cos   h)eˆ2 .
a* x  xa  q  0
(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 begin to develop the terms, a* x and xa
Figure 2 A Five-Jointed Robot
A slightly complicated application is to formulate
the kinematics for a five-jointed robot as shown
as follows.
a* x  (a0  ai eˆi )( x0  x j eˆ j )
 a0 x0  ai xi  (a0 xk  ak x0   ijk ai x j )eˆk
above.
Five pairs of quaternions encoding the rotations
xa  ( x0  x j eˆ j )(a0  ai eˆi )
 a0 x0  ai xi  (a0 xk  ak x0   ijk xi a j )eˆk
and translations are defined for performing a
sequence of affine transformations, i.e.
q5  cos
5
2
 sin
5
2
(21)
As a result of the indicial notion, the terms in
eˆ2 , b5   feˆ1  eeˆ2 , P 5  0 ,
q4  cos


4

 sin 4 eˆ1 , b4  deˆ2 , q3  cos 3  sin 3 eˆ1 , b3  ceˆ2 ,
2
2
2
2
q2  cos
2



 sin 2 eˆ3 , b2  beˆ2 , q1  cos 1  sin 1 eˆ3 , b1  aeˆ3 ,
2
2
2
2
(20), (21) are not only obtained in a way of great
efficiency but also in an algebraic transparency so
that we could easily render the following result.
a* x  xa  2a0 x0  2(a0 xk  2 ijk xi a j )eˆk
Hence,
0  a* x  xa  q
 2a0 x0  q0  [2(a0 xk   ijk xi a j )  qk ]eˆk
where a, b, c, d, e, f are the known length parameters
(22)
(23)
By the definition of the quaternion, a zero
of the linkages, and
quaternion implies that its scalar part and vector part
P 0  q1 (q2 (q3 (q4 (q5 P5 q5*  b5 )q4*  b4 )q3*
 b3 )q2*  b2 )q1*  b1
(20)
,
are all zero, i.e.
2a0 x0  q0  0
 P10eˆ1  P20eˆ2  P30eˆ3
2a0 xk  2 ijk xi a j  qk  0
where
e
P10  b sin 1  f cos(1   2 )  [sin(1   2  3   4 )
2
 sin(1   2  3   4 )]
d
 [sin(1   2  3 )  sin(1   2  3 )]  c sin(1   2 )
2
e
P20  b cos 1  f sin(1   2 )  [cos(1   2  3   4 )
2
 cos(1   2  3   4 )]
d
 [cos(1   2  3 )  cos(1   2  3 )]  c cos(1   2 )
2
.
(24)
Thus, the solution of (19) is obtained as follows.
x0 
x1 
x2 
x3 
q0
2a0
1
2a0 a
2
{(a02  a12 )q1  (a1a2  a0 a3 )q2  (a1a3  a0 a2 )q3 }
2
{(a1a2  a0 a3 )q1  (a02  a22 )q2  (a2 a3  a0 a1 )q3 }
2
{(a1a3  a0 a2 )q1  (a2 a3  a0 a1 )q2  (a02  a32 )q3 }
1
2a0 a
1
2a0 a
P30  e sin(3   4 )  d sin 3  a .
8. Conclusions
One additional example of solving a Lyapunov
like equation is illustrated below to justify the
It is tedious to write long expressions with lots
effectiveness of this approach. The Lyapunov
of components and imaginary units to manipulate the
equation often appears in the study of the stability of
quaternion.
a control system.
and the Einstein summation to simplify the algebra
Thus, we introduce the indicial notation
manipulation. In order to prevent the prodigious
growth in indicial notations, the dummy indices,
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 systems. 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, 1989.
[2] J. P. Ward, “Quaternions and Cayley Numbers,”
Kluwer Academic Publishers, London, 1997.
[3] Albert Einstein, “The Meaning 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., 1987.
[5] Janez Funda, Russell H. Taylor, Richard P. Paul
"On Homogeneous Transforms, Quaternions, and
Computational Efficiency," IEEE Transactions on
Robotics and Automation, Vol. 6, No. 3, pp.
382-388, June 1990.
[6] Richard D. Klafter, Thomas A. Chmielewski,
Michael
Negin,
"Robotic
Engineering
An
Integrated Approach," Prentice-Hall, Inc., New
Jersey, 1989.