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:
2Department of Applied Mathematics, Chung-HuaUniversity,
No. 707, Sec. 2, Wufu Rd., Hsinchu, 30012, Taiwan, R.O.C.
Tel:03-518-6392, Fax:03-537-3771
Email:
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
The quaternion [1][2], which is a generalization of a complex number was invented by Hamilton in 1843. He discovered that 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
. (1)
In Einstein’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 mechanics, 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 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. We use the to define the products of the imaginary units. 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 properties 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 glimpse of the efficiency of this approach as compared to the unabridgedapproach.
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 indices 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 avoid any possible confusion, we would like to make the following distinctions.
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 dummy variables. This convention could prevent the prodigious growth in indicial notations.
The following identity is extremely important and extensively used here.
(2)
where the Kronecker delta and Levi-Civita symbol are defined below.
3. Definition of Quaternion Algebra
Roughly speaking, algebra is a linear spaceover a field that admits a product operation. A precise definition is stated below.
Definition 1: Let S be a finite-dimensional linear space over a field F. If and,S is called an algebra if it possesses the following properties.
(i)
(ii)
We use the indicial notation to rewrite the representation of q as
(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 units stand for the base vectors,.To determine the multiplication rules, we assign the operations on the to be
. (4)
The addition rule of quaternion numbers is defined in terms of indicial notation.
(5)
The product of two quaternion numbers can be obtained by the rule for multiplying sums as follows.
(6)
Introducing (4) into the equation (6), we have
(7)
If the Equation (7) is unabridged, it is identical to the result obtained by the conventional approach, i.e.
. (8)
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.
(9)
Likewise, the rest can be proved to show that the quaternion indeed is an algebra.
4. Associative Normed Algebra
Another 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 an associative normed algebra, i.e.
(i)
(10)
(ii)
(11)
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.
The part (ii) is proved in a similar fashion. Using (7) to obtain , we have
and likewise. Hence,
The right hand side of identity in (ii) is
As we develop the quadratic terms of the above equation, it is very easy to identify thatand cancel each other. Hence,
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
where(ii) The right quotient of b by a is defined as
where The condition under which the quaternion is commutativeis co-linear on the vector part of the quaternions i.e..
Corollary 2-1: Inversion
Let q be a quaternion. Its inverse is equal to
(14)
Note thatis the conjugate of the quaternion and their vector parts are co-linear. Thus, there is no distinction between left inverse and right inverse of a quaternion.
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.
Lemma 3-1: Automorphism
Ifis a unit quaternion, i.e., a pure quaternionthrough automorphism induced by is equal to the following.
(15)
Proof:
Since q is a unit quaternion, it is obvious thatby corollary 2-1. In the sequel, we apply equation (7), identity and rearrange the dummy indices.
Q.E.D.
Supposed that a pure quaternionx rotates about a pure and unit quaternion p with a angle , its new position can be obtained by means of the automorphismas follows.
(16)
whereand for.
Theorem 3: Affine Transformation
Supposed thatis 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.
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 corollary, a consequence of theorem 3, states the conversion from an affine transformation to a homogeneous transformation.
Corollary 3-1: Homogeneous Transformation
Given specific quaternions, and, the affine transformation shown in theorem 3 can be represented by the corresponding homogeneous transformation
(18)
where
Proof:
Plunge in each component of q and b to the equation (17) and simplify them with the trigonometric functions, i.e.
Q.E.D.
As a result of the corollary 3-1, one translational transformation and three rotational transformations with respect to x,y,zcan be obtained as
,
,
,
.
7. Applications
In kinematics, the treatment of every problem is generally required to describe a motion with respect to the inertial frame. For instance, consider 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 horizontally.
Figure 1 A Broom Car System
In the aspect of homogeneous transformation, we need to define three reference frames attached to the system. The point at the tip of the rode is denoted asandto indicate its position with respect to frame-3 and inertial frame, respectively. Three homogeneous transformation matrices are defined as
, , .
Then, we have
where .
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.
, ,
The following result identical to the previous one is obtained using the affine transformation.
.
Figure 2 AFive-Jointed Robot
A slightly complicated application is to formulate the kinematics for a five-jointed robot as shown above.
Five pairs of quaternions encoding the rotations and translations are definedfor performing a sequence of affine transformations, i.e.
,
,
,
where a,b,c,d,e,fare the known length parameters of the linkages, and
,
where
.
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.
(19)
where are known quaternions but 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,and as follows.
(20)
(21)
As a result of the indicial notion, the terms in (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.
(22)
Hence,
(23)
By the definition of the quaternion, a zero quaternion implies that its scalar part and vector part are all zero, i.e.
. (24)
Thus, the solution of (19) is obtained as follows.
8. Conclusions
It is tedious to write long expressions with 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 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,”PrincetonUniversity 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.