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Quaternions and 3D Rotations

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Quaternions and 3D Rotations Quaternions and 3D Rotations Jeanette Schofield April 7, 2011 Contents 1 Quaternions: Definition and Basic Properties 3 1.1 Equality . 4 1.2 Addition . 4 1.2.1 Closed Under Addition . 4 1.2.2 Additive Identity . 4 1.2.3 Additive Inverse . 5 1.2.4 Commutative Under Addition . 5 1.2.5 Associative Under Addition . 5 1.3 Multiplication . 6 1.3.1 Scalar Multiplication . 6 1.3.2 Multiplication of Two Quaternions . 6 1.3.3 Associative Under Multiplication . 6 1.3.4 Implications of i2 = j2 = k2 = ijk = −1................. 7 1.3.5 Multiplication of Two Quaternions Revisited . 8 1.3.6 Closed Under Multiplication . 9 1.3.7 Multiplicative Identity . 9 1.3.8 Conjugate . 10 1.3.9 Norm . 10 1.3.10 Multiplicative Inverse . 10 1.4 Multiplication is Distributive over Addition . 11 1.5 Remarks on Quaternion Multiplication and Commutativity . 12 1.6 Summary of Algebraic Properties . 13 2 Quaternions and Rotations 14 2.1 Rotations in Three-Space . 14 2.2 Proof: w = qvq∗ ................................. 15 2.3 Expanding w = qvq∗ ............................... 20 2.4 Rotation Algorithm . 21 2.5 Examples . 22 2.5.1 Example 1 . 22 2.5.2 Example 2 . 23 3 Rotating a Coordinate Frame S into Another Coordinate Frame S0 26 3.1 Generalized Two Axis-Angle Rotation Method . 26 3.2 Two Axis-Angle Rotation Method Using Quaternions . 27 1 3.2.1 Example . 28 A Some Useful Equations 31 A.1 The Dot and Cross Product of Two Vectors . 31 A.1.1 The Dot Product . 31 A.1.2 The Cross Product . 31 A.2 Useful Relationships . 31 A.2.1 Proof: If v is ? to w, kv + wk2 = kvk2 + kwk2 . 32 B Source Code for the Two Axis-Angle Frame Rotation Method 33 B.1 C++ Code Listing . 33 2 Chapter 1 Quaternions: Definition and Basic Properties Quaternions may be thought of as 4-tuples of real numbers: q = (q0; q1; q2; q3). Equivalently, quaternions may also be written in the form q = q0 + q1i + q2j + q3k (1.0.0.1) where i = (0; 1; 0; 0) j = (0; 0; 1; 0) (1.0.0.2) k = (0; 0; 0; 1) are themselves 4-tuples which satisfy the following relations i2 = j2 = k2 = ijk = −1: (1.0.0.3) 3 In R , a vector is defined as v = vxi + vyj + vzk where i, j, and k are vectors of the form i = (1; 0; 0) j = (0; 1; 0) k = (0; 0; 1): 3 4 Any vector in R can be mapped to R by (vx; vy; vz) 7! (0; vx; vy; vz) This mapping is often represented by v = 0 + v = v. By convention, the i, j, and k components of a quaternion are usually written as vectors. Thus, any quaternion q may be written as q = q0 + q = q0 + q1i + q2j + q3k (1.0.0.4) where i2 = j2 = k2 = ijk = −1: (1.0.0.5) Here, q0 is referred as the scalar part of the quaternion and q = q1i + q2j + q3k is referred to 3 as the vector part of the quaternion. The quaternions form a non-commutative division ring. This means that quaternions have all the properties of a field except that quaternions are not commutative under multi- plication. A discussion of the various properties of quaternions follows. 1.1 Equality Two quaternions, p = p0 + p1i + p2j + p3k and q = q0 + q1i + q2j + q3k are considered equal if and only if their corresponding scalar components are equal: p0 = q0 p1 = q1 p2 = q2 p3 = q3: 1.2 Addition Two quaternions, p = p0 + p1i + p2j + p3k and q = q0 + q1i + q2j + q3k, can be added together by combining their corresponding scalar and vector components: p + q = (p0 + p1i + p2j + p3k) + (q0 + q1i + q2j + q3k) = (p0 + q0) + (p1 + q1)i + (p2 + q2)j + (p3 + q3)k: The following subsections discuss various properties of the quaternions under addition. 1.2.1 Closed Under Addition The addition of two quaternions, p = p0 + p1i + p2j + p3k and q = q0 + q1i + q2j + q3k, will result in a third quaternion, r = r0 + r1i + r2j + r3k: p + q = p0 + p1i + p2j + p3k + q0 + q1i + q2j + q3k = (p0 + q0) + (p1 + q1)i + (p2 + q2)j + (p3 + q3)k = r0 + r1i + r2j + r3k = r: 1.2.2 Additive Identity The additive identity for quaternions is the zero quaternion, 0 = 0 + 0i + 0j + 0k: q + 0 = q0 + q1i + q2j + q3k + 0 + 0i + 0j + 0k = (q0 + 0) + (q1 + 0)i + (q2 + 0)j + (q3 + 0)k = q0 + q1i + q2j + q3k = q: 4 1.2.3 Additive Inverse The additive inverse for a quaternion q = q0 + q1i + q2j + q3k is given by −q = −q0 − q1i − q2j − q3k: q + (−q) = (q0 + q1i + q2j + q3k) + (−q0 − q1i − q2j − q3k) = (q0 − q0) + (q1 − q1)i + (q2 − q2)j + (q3 − q3)k = 0: 1.2.4 Commutative Under Addition Commutativity means that the order in which two quaternions are added together does not matter. Thus, for any two quaternions, p = p0 + p1i + p2j + p3k and q = q0 + q1i + q2j + q3k, p + q = (p0 + p1i + p2j + p3k) + (q0 + q1i + q2j + q3k) = (p0 + q0) + (p1 + q1)i + (p2 + q2)j + (p3 + q3)k = (q0 + p0) + (q1 + p1)i + (q2 + p2)j + (q3 + p3)k = (q0 + q1i + q2j + q3k) + (p0 + p1i + p2j + p3k) = q + p: 1.2.5 Associative Under Addition Associativity means given any three quaternions under addition, the order in which they are grouped together does not matter. For any three quaternions, p = p0 + p1i + p2j + p3k, q = q0 + q1i + q2j + q3k, and r = r0 + r1i + r2j + r3k, observe that (p + q) + r = (p0 + q0) + (p1 + q1)i + (p2 + q2)j + (p3 + q3)k + r0 + r1i + r2j + r3k = (p0 + q0) + r0 + (p1 + q1) + r1 i + (p2 + q2) + r2 j + (p3 + q3) + r3 k: Because the real numbers are associative under addition, the scalar componenets can be manipulated. Thus, (p + q) + r = p0 + (q0 + r0) + p1 + (q1 + r1) i + p2 + (q2 + r2) j + p3 + (q3 + r3) k = p0 + p1i + p2j + p3k + (q0 + r0) + (q1 + r1)i + (q2 + r2)j +(q3 + r3)k = p + (q + r): To summarize, the associativity of quaternions under addition means that the order in which three or more quaternions are combined does not matte; i.e. (p + q) + r = p + (q + r). 5 1.3 Multiplication For quaternions, there are two types of multiplication to discuss: scalar multiplication and the muiltiplication of two quaternions. Scalar multiplication involves the multiplication of a quaternion q = q0 +q1i+q2j+q3k by any scalar s 2 R. The multiplication of two quaternions involves multiplying two quaternions p = p0 + p1i + p2j + p3k and q = q0 + q1i + q2j + q3k. Various properties of quaternions that deal with multiplication will also be discussed in this section. 1.3.1 Scalar Multiplication Multiplication of a quaternion q = q0 + q1i + q2j + q3k by a scalar s is defined as sq = sq0 + sq = sq0 + sq1i + sq2j + sq3k: 1.3.2 Multiplication of Two Quaternions The multiplication of two quaternions, p = p0 + p and q = q0 + q, is defined as pq = (p0 + p)(q0 + q) = p0q0 + p0q + q0p + pq: (1.3.2.1) At this point is in unclear how to expand the term "pq." This will be revisted in a later section. 1.3.3 Associative Under Multiplication Quaternions are associative under multiplication. This means that for any three quaternions, p = p0 + p1i + p2j + p3k, q = q0 + q1i + q2j + q3k, and r = r0 + r1i + r2j + r3k, the order in which the quaternions are grouped together under multiplication does not matter: (pq)r = (p0q0 + p0q + q0p + pq)(r0 + r) = p0q0r0 + p0q0r + p0r0q + p0qr + q0r0p + q0pr + r0pq + pqr: Factoring p0 out of the first four terms and p out of the last four terms, (pq)r = p0(q0r0 + q0r + r0q + qr) + p(q0r0 + q0r + r0q + qr) = (p0 + p)(q0r0 + q0r + r0q + qr): By definition, qr = q0r0 + q0r + r0q + qr. Thus, (pq)r = p(qr) (1.3.3.1) and it has been shown that quaternions are associative under multiplication. 6 1.3.4 Implications of i2 = j2 = k2 = ijk = −1 The relationships i2 = j2 = k2 = ijk = −1 must always hold; some important relationships that follow from these statements will now be derived. Since i2 = ii = ijk = −1, this implies that i = jk: (1.3.4.1) Similarly, since k2 = kk = ijk = −1, then k = ij: (1.3.4.2) Using equation (1.3.4.2) and multiplying by j on the right, k(j) = ij(j) kj = i(jj) = i(j2) = i(−1) = −i so i = −kj: (1.3.4.3) From equations (1.3.4.1) and (1.3.4.3), the following relationships can be found for i: i = jk = −kj: Using equation (1.3.4.1) and multiplying on the left by j, (j)i = (j)jk ji = (jj)k = (j2)k = (−1)k = −k so k = −ji: (1.3.4.4) From equation (1.3.4.2) and (1.3.4.4), the following relationships are found for k: k = ij = −ji: Using equation (1.3.4.4) and multiplying on the right by i, k(i) = −ji(i) ki = −j(ii) = −j(i2) = −j(−1) = j: Thus, j = ki: (1.3.4.5) Using equation (1.3.4.2) and multiplying on the left by i, (i)k = (i)ij 7 ik = (ii)j = (i2)j = (−1)j = −j: Hence, j = −ik: (1.3.4.6) From equations (1.3.4.5) and (1.3.4.6), the following relationships are found for j: j = ki = −ik: To summarize, when multiplying two quaternions together, the relationships i2 = j2 = k2 = ijk = −1 must always hold.
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