1 Quaternion Computation Neil Dantam [email protected] Institute for Robotics and Intelligent Machines, Georgia Institute of Technology, Atlanta, GA, USA Work in Progress – Updated 2014-10-01. IV-F Conjugate................6 IV-G Exponential...............6 Abstract—Quaternions are a useful representation for orienta- tion, and dual quaternions extend the representation to handle IV-H Logarithm................7 translations as well. This report discusses computations that can IV-I Chaining Transforms..........7 be performed using quaternions. To accurately compute results IV-J Transforming a point..........7 near singularities, we provide Taylor series approximations which can be efficiently computed to within machine precision. IV-K Derivatives................7 IV-K1 Product Rule.........7 IV-K2 Angular Velocity.......7 CONTENTS IV-K3 Translational Velocity....7 I Introduction 1 IV-L Integration................7 I-A Notation.................2 V Implicit Dual Quaternions 7 II Quaternions 2 V-A Chaining transforms...........7 II-A Representation..............2 V-B Transforming points...........8 II-B Multiplication..............2 V-C Conjugate................8 II-B1 Cross and dot product defi- V-D Derivatives................8 nition.............2 II-B2 Matrix definition.......2 VI Matrices and Euclidean Transforms 8 II-B3 Properties..........3 VI-A Rotation Matrix.............8 II-B4 Pure Multiplication.....3 VI-B Transformation Matrix.........8 II-C Norm...................3 VI-C Transforming Points...........8 II-D Conjugate................3 II-E Inverse..................3 VI-D Chaining Transforms..........8 II-F Exponential...............3 II-G Logarithm................3 References 8 II-H Power..................4 II-I Pure Exponential Derivative.......4 Appendix A: History 8 II-J Unit Logarithm Derivative.......4 II-K Unit Quaternion Angle.........4 Appendix B: Derivation of Quaternion Multiplication 9 II-L Product Rule...............4 B-A Derivation of Quaternion Basis Equalities9 B-B Derivation of Quaternion Multiplication9 III Representing Orientation 4 III-A Rotating a vector............5 III-B Chaining rotations............5 I. INTRODUCTION III-C Angular Derivatives...........5 III-D Axis-Angle................5 Quaternions are a convenient representation for spatial mo- III-E Spherical Linear Interpolation.....5 tion that provides some computational advantages over other III-F Integration................5 methods. III-G Finite Difference.............6 The straightforward definitions of many quaternion quan- tities, particularly exponentials, logarithms, and derivatives, IV Dual Quaternions and Euclidean Transforms 6 contain singularities where a denominator goes to zero. We can IV-A Dual Numbers..............6 avoid computational problems at these points by computing IV-B Representation..............6 key factors near the singularity using a Taylor series, though IV-C Construction...............6 this may require some careful rearrangement of terms to IV-D Multiplication..............6 identify suitable factors and series. IV-E Matrix Form...............6 A Taylor series evaluated near point a is: Quaternions Associative p ⊗ (q ⊗ r) = (p ⊗ q) ⊗ r Distributive p ⊗ (q + r) = p ⊗ q + p ⊗ r 1 (n) NOT Commutative p ⊗ q 6= q ⊗ p X f (a) ∗ ∗ ∗ f(x) = (x − a)n Conjugate Mul. (p ⊗ q) = q ⊗ p n! Conjugate Add. (p + q)∗ = q∗ + p∗ n=0 TABLE I f 0(a) f 00(a) ALGEBRAIC QUATERNION PROPERTIES = f(a) + (x − a) + (x − a)2 + 1! 2! f (3)(a) (x − a)3 + ::: (1) 3! A. Representation To evaluate the infinite series to machine precision, we only We represent a quaternion as a 4-tuple of real numbers: need to compute up the term below floating point round-off. The resulting approximation is a polynomial which can q = w + xi + yj + zk be efficiently evaluated using Horner’s Rule, Algorithm1. = (x y z w) f(n)(a) The coefficients are the terms n! and the indeterminate = H(qv; w) (7) variable is x − a. Note than many Taylor series have zero coefficents for the odd or even terms. We can produce a more Historically, qv is called the vector part of the quaternion compact Horner polynomial by omitting the zero coefficients, and qw the scalar part. using (x − a)2 as the indeterminate variable, and perhaps It is convenient to define quaternion operations in terms of multiplying the whole result by (x − a). vector and matrix operations, so we also the whole quaternion as a column vector. This also provides an in-memory storage representation. Algorithm 1: Horner’s Rule T Input: b0; b1; : : : bn : Coefficients ~q = [x y z w] (8) Input: z : Indeterminate Variable T ~qv = [x y z] (9) Output: y : Result 1 y bn A alternate convention stores terms in wxyz order, so when 2 y bn−1 + zy using different software packages, it is sometimes necessary 3 y bn−2 + zy to convert between orderings. 4 ... 5 y b0 + zy B. Multiplication From the definition of the basis elements (2), we obtain a formula for quaternion multiplication. See section B for the detailed derivation. A. Notation 1) Cross and dot product definition: We define quaternion We adopt the following abbreviations to condense notation: multiplication in terms of cross products and dot products of its elements: • Quaternions are typeset as q. • Dual Quaternions are typeset as S. ~qv × ~pv + qw~pv + pw~qv • Vectors are typeset as ~x. q ⊗ p = (10) qwpw − ~qv · ~pv • Matrices are typeset as A. • Time derivatives of variable x are given as x_. 2) Matrix definition: Expanding the above terms, we can • Sines and cosines are abbreviated as s and c. express quaternion multiplication as matrix multiplication: q ⊗ p = 2 3 II. QUATERNIONS qw −qz qy qx q q −q q Quaternions are an extension of the complex numbers, using Q ~p = 6 z w x y 7 ~p = L 6−q q q q 7 basis elements i, j, and k defined as: 4 y x w z 5 −qx −qy −qz qw 2 2 2 2 3 i = j = k = ijk = −1 (2) pw pz −py px 6−pz pw px py 7 From (2), it follows: PR~q = 6 7 ~q = 4 py −px pw pz 5 −px −py −pz pw jk = −k j = i (3) 2 3 qxpw + qypz + qwpx − qzpy ki = −ik = j (4) 6 qzpx + qwpy + qypw − qxpz 7 6 7 (11) ij = −ji = k (5) 4 qwpz + qzpw + qxpy − qypx 5 −(qypy + qxpx + qzpz − qwpw) A quaternion, then, is: This matrix form is more suitable for efficient implementa- q = w + xi + yj + zk (6) tion computation using SIMD instructions. 2 Quaternions 3) Properties: Quaternion multiplication is associative and I distributive, but it is not commutative. 4) Pure Multiplication: When multiplying by a pure quater- nion, i.e., zero scalar part, we can simplify: 2 3 qw −qz qy q q −q jq j ⊗ (v; 0) = 6 z w x7 v = v q 6−q q q 7 4 y x w 5 φ −qx −qy −qz 2 q v 3 2q v 3 2 q v 3 R y z z y w x qw q v q v q v 6 z x 7 − 6 x z7 + 6 w y 7 (12) 6 q v 7 6q v 7 6 q v 7 4 x y 5 4 y x5 4 w z 5 Fig. 1. Imaginary Plane for Quaternions −qxvx qyvy −qzvz 2 3 qw qz −qy 6−qz qw qx 7 some useful trigonometric ratios for analyzing quaternion (v; 0) ⊗ q = 6 7 v = 4 qy −qx qw 5 functions: −qx −qy −qz 2 3 2 3 2 3 φ = atan2 (jqvj ; qw) (19) vyqz vzqy vxqw jqvj 6 vzqx 7 6vxqz7 6 vyqw 7 sin (φ) = (20) 6 7 − 6 7 + 6 7 (13) jqj 4 vxqy 5 4vyqx5 4 vzqw 5 qw −vxqx vyqy −vzqz cos (φ) = (21) 2 3 jqj uyvz − uzvy 6 uzvx − uxvz 7 u × v The quaternion exponential is: (u; 0) ⊗ (v; 0) = 6 7 = (14) 4 uxvy − uyvx 5 −u · v sin (jq j) q qw v −uxvx − uyvy − uzvz e = e H qv ; cos (jqvj) (22) jqvj Thus, the case of multiplying two pure quaternions simpli- fies to the commonly used cross (×) and dot (·) products. When jqvj approaches zero, we can use the Taylor series approximation: C. Norm sin (θ) θ2 θ4 θ6 = 1 − + − + ::: (23) p θ 6 120 5040 jqj = ~q · ~q (15) For a pure quaternion, the exponential simplifies to: A unit quaternion has norm of one. ( q sin(jqv j) e = H qv ; cos (jqvj) jqv j D. Conjugate qw = 0 =) (24) jeq j = 1 ∗ q = H(−qv; qw) (16) G. Logarithm E. Inverse To compute the logarithm, first consider the angle between the vector and scalar parts of the quaternion. q∗ q−1 = (17) ~q · ~q q jq j φ = cos−1 w = sin−1 v = atan2 (jq j ; q ) (25) Note that for unit quaternions, the inverse is equal to the jqj jqj v w conjugate. The atan2 form to compute φ is generally best for numerical F. Exponential stability. The exponential shows the relationship between quaternions and complex numbers. Recall Euler’s formula for complex φ ln q = H qv; ln (jqj) (26) numbers: jqvj φ iθ When jqvj approaches zero, we can compute as follows: e = cos (θ) + i sin (θ) (18) jqv j which relates the exponential function with angles in the φ φ φ φ complex plane. Similarly for quaternions, we can consider the = jqj = jqj = sin(φ) jq j (27) jqvj v sin (φ) jqj angle between the real and imaginary parts, Figure 1, yielding jqj 3 Representing Orientation φ Representation Storage Then, sin(φ) can be approximated by Taylor series: Quaternion 4 Axis-Angle 4 θ θ2 7θ4 31θ6 Rotation Vector 3 = 1 + + + + ::: (28) Euler Angles 3 sin (θ) 6 360 15120 Rotation Matrix 9 TABLE II For a unit quaternion, the logarithm simplifies to: STORAGE REQUIREMENTS FOR ORIENTATION REPRESENTATIONS φ q = 1 =) ln(q) = H qv; 0 (29) Representation Chain Rotate Point sin (φ) Quaternion 16 multiply, 12 add 15 multiply, 15 add Rotation Matrix 27 multiply, 18 add 9 multiply, 6 add TABLE III H.