1 Historical Notes. Kinematics 5: Quaternions

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1 1 Historical Notes Quaternions were invented by the Irish mathematician William Rowan Hamilton in the late 1890s. The story goes 1 that Hamilton has pondered the problem of dividing one vector by another for years and the answer came to him in a flash of insight. The essence of the required properties of quaternions: 1 i 2 j 2 k 2 ijk 1 was supposedly scratched into a bridge in Dublin lest it be forgot. Quaternions share all of the properties of the real and complex numbers with the exception of the commutativity of multiplication. They can be used to represent affine transformations and projections - everything that a Homogeneous transform can represent. 1. There is always a story. Kalman supposedly invented his famous filter on a European train in a flash of insight.

2 1.1 Why Use Em? They are the only way to solve some problems - like the problem of generating regularly spaced 3D angles. They are the simplest way to solve other problems. Some problems in registration for example, can be solved in closed form. They are the fastest way to solve some problems. The quaternion loop in an inertial navigation system updates vehicle attitude 1000 times a second. 2 Representations and Notation 2 1.1Why Use Em? 2 Representations and Notation More than half of the confusion about quaternions is caused by all the different notations in use. Hamilton liked to represent quaternions as 4- tuples like so: ( q 0,,, ) Quaternions can also be viewed as hypercomplex numbers 1 with one real and 3 imaginary parts: q q 0 + i+ j+ k where i,j,k are called the principle imaginaries or quaternionic units. In this form, they can be manipulated as if they were polynomials in the variables i,j,k. 1. Even longer hypercomplex numbers like octernions have been defined. Generally in 1884, H. G.Grassman defined: q ae 1 + be 2 + ce 3 + de 4 + as a hypercomplex number where the e i s are the fundamental units. If these objects are to be treated like polynomials and the product of two such objects is to remain such an object, then a multiplication table similar to the quaternion table is necessary.

3 Compare this with standard complex numbers of the form: q q 0 + i Alternately, some people write them as the sum 1 of a traditional scalar and a traditional 3-vector: q q+ q where: q q 0 and: q i+ j+ k 2 Representations and Notation 3 1.1Why Use Em? Others write them as an ordered pair of real and complex parts: ( qq, ) By analogy to the exponential notation of complex numbers z e iθ, mathematicians have often used exponential notation. Thus: q e means the quaternion for a rotation by the angle θ about the axis w θw 1. What does it mean to add the two elements a and bi?. It means close to nothing. This sum is the same sum notation used in complex numbers and polynomials. It means these two things are treated together as a unit. The number a + bi is just notation for the ordered pair ( ab, ) and we will never ask for a + bi to be simplified by actually adding these things together. The sum is considered to the simplest form because the components themselves cannot be added. The sum does, however, have a function when the law of distribution is invoked.

4 3 Properties Quaternions are elements of a vector space which is endowed with multiplication 1. By this we mean that the product of two quaternions is defined. Ideally, we would like the product of two quaternions to be a quaternion. If we also want to be able to treat them as polynomials in i,j,k, then an expression like: p q ( p 0 + p 1 i+ p 2 j + p 3 k) ( q 0 + i + j+ k) 3 Properties 4 1.1Why Use Em? multiplies out to be the sum of all the elements in the following table: Table 1: Quaternion Multiplication q 0 i j k p 0 p 0 q 0 p 0 i p 0 j p 0 k p 1 i p 1 q 0 i p 1 i 2 p 1 ij p 1 ik p 2 j p 2 q 0 j p 2 ji p 2 j 2 p 2 jk p 3 k p 3 q 0 k p 3 ki p 3 kj p 3 k 2 i 2 For this to be a quaternion, the terms like and jk must somehow turn into scalars or singletons in the imaginaries like k. In fact, the rules that apply are as follows: Table 2: Quaternion Multiplication Table i j k i -1 k -j 1. There are at least 5 types of products defined on vectors. Scalar multiplication is defined so that kv changes the length of the vector v but not its direction. The dot (or scalar ) product a b produces a scalar from two vectors. The vector (or cross ) product a b produces a vector from two vectors. The outer product is the matrix product ab T. The quaternion product is the fifth. It produces a quaternion from two quaternions so it is similar to the vector cross product. j -k -1 i k j -i -1

5 and these are a consequence of the more compact rule: i 2 j 2 k 2 ijk 1 The table is easy to remember. The offdiagonal elements are the same rules associated with the vector cross product and the diagonal elements are the extension of the rule for complex numbers 1. It is convenient at different times to visualize quaternions in different ways. It is useful to switch back and forth between viewing them as: complex numbers 4-vectors in matrix algebra. the sum of a scalar and a 3-vector polynomials in the variables i,j,k Quaternions are generalizations of familiar number systems because the reals and the complex numbers are embedded in them. By 1. Self multiplication of the principle imaginaries does not follow the rules for the cross product because i i 0 whereas ii 1. 3 Properties 5 3.1Quaternion Product this we mean that a scalar and a complex number are special cases of quaternions and legitimate forms of quaternions. 3.1 Quaternion Product Ways to write the product of two quaternions depend on the notation used but all are equivalent. In hypercomplex number form: p q ( p 0 + p 1 i + p 2 j + p 3 k) ( q 0 + i + p 2 j + p 3 k) p q ( p 0 q 0 p 1 p 2 p 3 ) + ( )i + In scalar-vector form, this same result can be written as follows: p q ( p + p) ( q+ q) p q pq + pq + qp + pq The first three terms are easy to interpret. The last term can be written in terms of both the dot and cross products: pq p q p q

6 Thus, the quaternion product can be thought of as a superset of these operations. It is convenient to summarize this result like so: p q pq p q + pq + qp + p q Because the cross product does not commute, the quaternion product also does not commute. In general: p q q p 3.2 Quaternion Addition Quaternion addition follows the same pattern as complex numbers, vectors, and polynomials - we add them element by element: p + q ( p 0 + q 0 ) + ( p 1 + )i + ( p 2 + )j + ( p 3 + )k 3 Properties 6 3.2Quaternion Addition 3.3 Distributivity A very important property is that of distributivity: ( p + q )r p r + q r p ( q + r ) p q + p r Many derivations involving quaternions become easy due to this property. 3.4 Quaternion Dot Product & Norm We can define a dot product on quaternions (not to be confused with the quaternion product): p q pq + p q which is just the sum of pairwise products as if we were taking the norm of a 4 dimensional vector. Given this, the quaternion norm (equivalent of length) is defined as: q q q

7 3.5 Unit Quaternions A quaternion whose norm is 1 is called a unit quaternion. Note that the product of two unit quaternions is a unit quaternion so unit quaternions constitute a subgroup of general quaternions. 3.6 Quaternion Conjugate As for complex numbers, the conjugate of a quaternion is formed by reversing the sign of the complex part: q * q q Using the product equation, several things cancel in the quaternion product with the conjugate leaving: q q * ( qq + q q) q q which is the same result as the dot product. Thus, we have a second way to get the norm of a quaternion: q q q * 3 Properties 7 3.5Unit Quaternions 3.7 Quaternion Inverse More importantly this is now a quaternion product (not a dot product) which produces a scalar. Since scalars are perfectly legitimate quaternions, this is OK. Don t assume that a quaternion must have vector or complex part. The multiplicative inverse of a quaternion (or anything else) is the thing which, when multiplied by it, produces unity: q q 1 1 Since: q q * q 2 1 we have the multiplicative inverse of a quaternion as 1 : q 1 q * q 2 1. This result is huge. Remember the question that Hamilton pondered for years was how do I divide a 3 vector by another? An equivalent question is What is the (multiplicative) inverse of a quaternion? We have the answer here.

8 3.8 Conjugate of the Product The product of two conjugates is the same as conjugate of the product in the opposite order: p *q * ( p p) ( q q) ( pq + p q) + ( pq qp + p q) ( q p ) * [( q+ q) ( p + p) ] * [( qp + q p) + ( qp + pq + q p) ] * [( qp + q p) + ( qp pq q p) ] ( pq + p q) + ( pq qp + p q) That is: ( q p ) * p *q * This is useful is several proofs. 4 Representing 3D Rotations 8 3.8Conjugate of the Product 4 Representing 3D Rotations 3D rotations can be represented conveniently in terms of unit quaternions. The quaternion: θ θ q cos-- + ŵsin can be interpreted to represent a rotation operator in 3D by the angle θ about the (unit vector) axis ŵ. Note that ŵ is now being interpreted as a vector in 3D rather than a hypercomplex number. That was the conversion from axis-angle ( θ, ŵ) notation to quaternion. Clearly, the inverse is: θ 2atan2( q, q) ŵ q q Taking the negative q represents the same rotation (opposite direction about the negative

9 vector) whereas taking the conjugate q * produces the inverse rotation. Let us quaternize a vector x by making it the vector part of an associated quaternion: x 0 + x This vector can be operated upon by a unit quaternion thus: x ' q x q * and this will have a zero scalar part like the original x. Finally, the punch line is this. Because rotation is accomplished by a quaternion product, and because quaternion products are associative, it follows that compound rotation operations are formed by multiplying the two unit quaternions: x '' p x ' p * ( p q )x ( q *p * ) 4 Representing 3D Rotations 9 4.1Rotation Matrix Equivalent product ( p q ) rotates a vector first by q and then by p. 4.1 Rotation Matrix Equivalent Of course, we know that a matrix can also implement a general 3D rotation operator. Therefore, there must be a way to compute the matrix from the quaternion and vice versa. For the quaternion: q q 0 + i+ j+ k the equivalent 3D rotation matrix is: R 2 2 2[ q 0 + ] 1 2[ q 0 ] 2[ + q 0 ] 2 2 2[ + q 0 ] 2[ q 0 + ] 1 2[ q 0 ] 2 2 2[ q 0 ] 2[ q 0 + ] 2[ q 0 + ] 1 This is tedious but easy to show if you are organized in your algebra by expanding q x q * and collecting like terms. Using the earlier result for the product of two conjugates, we have shown that the

10 The inverse problem is also solvable. Suppose we have the matrix: R r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 Can we equate this to the last result and solve for the quaternion elements? You bet. Notice that: r 32 r 23 2q 0 r 32 + r 23 2 r 13 r 31 2q 0 r 13 + r 31 2 r 21 r 12 2q 0 r 21 + r 12 2 If one of the q i is known, the rest can be determined from these equations. We can find 4 Representing 3D Rotations Rotation Matrix Equivalent one q i from combinations of the diagonal elements: 2 r 11 + r 22 + r 33 4q r 11 r 22 r 33 4 r 11 r 22 r 33 These equations assume the original quaternion was a unit quaternion but the general solution is not much harder to derive. In practice any of these can be zero, so the largest of these is computed and used to find the others using the earlier equations. We determine only one q i using the second set to get the signs right. The sign of any one q i can be assigned arbitrarily since the negative of a quaternion represents the same rotation r 11 r 22 + r

11 5 Summary Quaternions are hypercomplex numbers which are useful in practice for modelling 3D rotations. There are several notations in use and several ways to interpret what they mean. Its helps to be able to switch back and forth between all of them. There are conversion formulas between quaternion and matrix representations of 3D rotations. 5 Summary Rotation Matrix Equivalent 6 References [1] BKP Horn, Robot Vision, MIT Press / McGraw-Hill, pp [2] E Pervin, and J Webb, Quaternions in Computer Vision and Robotics, CMU-CS [3] H. Eves, Foundations and Fundamental Concepts of Mathematics, 3rd Ed. Dover. [4] W. R. Hamilton, Elements of Quaternions, Chelsea, New York, 1969.

12 7 Notes Rotation Matrix Equivalent 7 Notes Add something on democracy of unit vectors. See Pervin. Add something on how translations can be done by quaternions as well. Do an example of an INS quaternion loop.