Quaternions and their applications

Transcription

1 Quaternions and their applications Ramachandran Subramanian February 20, 2014

2 Jack B. Kuipers. Quaternions and Rotation Sequences. university press, New Jersey, 1998 Princeton

3 Outline 1 Background 2 Introduction 3 Review of rotation operations - matrix approach 4 Rotation operations - quaternion approach 5 An example 6 Conclusion

4 Similarity between complex numbers and 2D vectors The set of two numbers (1, 1) can be used to represent either a vector a = 1ˆ ı + 1ˆ j or a complex number A = 1 + 1ˆ ı Addition of two complex numbers A(1, 1) and B(0, 1) is similar to addition of two vectors a(1, 1) and b(0, 1) Multiplication of two complex numbers is similar to rotation of a vector about an axis passing through the origin e.g., C = A B = 1 + 1ˆ ı (Note: arg(b) = 90 ) 2D vectors : complex numbers :: 3D vectors :? This problem troubled William Rowan Hamilton for many years 1 of 21

5 The solution He tried using sets of three numbers (triplets) to achieve similar properties to 3D vectors In the case of multiplication of 2 complex numbers, the corresponding rotation of the 2D vector was always about the z-axis passing through the origin The first problem he faced with the rotation of 3D vectors was finding the axis about which the rotation would take place (if he multiplied 2 triplets) The second problem was there was no easy way to associate an angle to a triplet In 1843, when he was walking along the Royal canal, he was struck with this sudden idea that he needed sets of four numbers to solve all his problems He etched his famous rule on a nearby bridge: ı 2 = j 2 = k 2 = ı jk = 1 (1) 2 of 21

6 Definition of a quaternion A quaternion is defined as: q = q 0 + q i.e., a scalar + a vector!!! (where q = q 1ˆ ı + q 2ˆ j + q 3ˆk) We need to define further rules for quaternion operations. Let p = p 0 + p 1ˆ ı + p 2ˆ j + p 3ˆk and q = q0 + q 1ˆ ı + q 2ˆ j + q 3ˆk be two quaternions Equality: Addition: p = q p 0 = q 0, p 1 = q 1, p 2 = q 2, p 3 = q 3 (2) q + p = (q 0 + p 0 ) + (q 1 + p 1 )ˆ ı + (q 2 + p 2 )ˆ j + (q 3 + p 3 )ˆk (3) Multiplication by a scalar: cp = cp 0 + cp 1ˆ ı + cp 2ˆ j + cp 3ˆk (4) Multiplication of two quaternions is more complicated 3 of 21

7 Multiplication of two quaternions p = p 0 + p 1ˆ ı + p 2ˆ j + p 3ˆk and q = q0 + q 1ˆ ı + q 2ˆ j + q 3ˆk ı 2 = j 2 = k 2 = ı jk = 1 From Hamilton s famous rule (eq. (1)),it follows: ı j = k = ı j; jk = ı = k j; k ı = j = ık; (5) The quaternion product pq is defined as: pq =p 0 q 0 (p 1 q 1 + p 2 q 2 + p 3 q 3 ) + p 0 (q 1ˆ ı + q 2ˆ j + q 3ˆk) + q0 (p 1ˆ ı + p 2ˆ j + p 3ˆk) + ˆ ı(p 2 q 3 p 3 q 2 ) + ˆ j(p 3 q 1 p 1 q 3 ) + ˆk(p 1 q 2 p 2 q 1 ) =p 0 q 0 p q + p 0 q + q 0 p + p q (6) 4 of 21

8 Complex conjugate, norm and inverse Complex conjugate: q = q 0 q; q + q = 2q 0 (7) Norm: Inverse: N(q) = q q = q 20 + q21 + q22 + q23 = q 2 (8) qq 1 = 1 q 1 = q q 2 (9) 5 of 21

9 Rotations in R 3 A few points to keep in mind with regards to rotations in 3 dimensions: Length of the vector remains unchanged Angle between two vectors and hence, the dot product also remain unchanged The axis of rotation itself remains unchanged as a result of the rotation Two perspectives of rotation: 1) Rotation of vector and 2) Rotation of reference frame Both perspectives yield exactly the same result but have slightly different representations 6 of 21

10 Rotation of reference frame Let P be a point whose coordinates are (x 1, y 1, z 1 ) relative to the XYZ coordinate frame We rotate the frame about the the Z axis by an angle θ Let (x 2, y 2, z 2 ) be the coordinates relative to the rotated frame Since rotation was about the Z axis, z 2 = z 1 Based on simple geometry, we can write down the following: x 2 = x 1 cos θ + y 1 sin θ y 2 = y 1 cos θ x 1 sin θ Combining all the 3 equations, we get: x 2 cos θ sin θ 0 x 1 y 2 = sin θ cos θ 0 y 1 (10) z z 1 7 of 21

11 Rotation matrix or operator Given any vector v 1 relative to XYZ coordinate frame and a vector v 2 relative to a rotated frame, the rotation matrix or operator is given as v 2 = A v 1 The rotation matrix A is orthonormal i.e., AA t = 1 and A = 1 If A 1, A 2 and A 3 represent three successive rotations,then A = A 3 A 2 A 1 represents a composite rotation Axis of the composite rotation is the eigenvector of A whose eigenvalue is 1 Angle associated with A is given by θ = cos 1 [ Tr(A) 1 2 It might be a bit tedious to construct the rotation matrix geometrically for arbitrarily chosen axis of rotation ] (11) 8 of 21

12 Questions about quaternions q = q 0 + q = q 0 + q 1ˆ ı + q 2ˆ j + q 3ˆk Can quaternions be used as alternative rotation operators? How can quaternions R 4 operate on vectors R 3 How to associate an axis of rotation with the quaternion? How to associate an angle with the quaternion? Note: We have not yet formally defined the operator and all this is speculation 9 of 21

13 Questions about quaternions q = q 0 + q = q 0 + q 1ˆ ı + q 2ˆ j + q 3ˆk Yes How can quaternions R 4 operate on vectors R 3 How to associate an axis of rotation with the quaternion? How to associate an angle with the quaternion? Note: We have not yet formally defined the operator and all this is speculation 9 of 21

14 Questions about quaternions q = q 0 + q = q 0 + q 1ˆ ı + q 2ˆ j + q 3ˆk Yes Treating vectors as pure quaternions i.e. q 0 = 0 How to associate an axis of rotation with the quaternion? How to associate an angle with the quaternion? Note: We have not yet formally defined the operator and all this is speculation 9 of 21

15 Questions about quaternions q = q 0 + q = q 0 + q 1ˆ ı + q 2ˆ j + q 3ˆk Yes Treating vectors as pure quaternions i.e. q 0 = 0 Axis of rotation u = q/ q How to associate an angle with the quaternion? Note: We have not yet formally defined the operator and all this is speculation 9 of 21

16 Questions about quaternions q = q 0 + q = q 0 + q 1ˆ ı + q 2ˆ j + q 3ˆk Yes Treating vectors as pure quaternions i.e. q 0 = 0 Axis of rotation u = q/ q q = 1 or q q 2 = 1 q = cos θ + u sin θ Note: We have not yet formally defined the operator and all this is speculation 9 of 21

17 Constructing the quaternion rotation operator Trial and error: Let v and w be vectors relative to the XYZ coordinate frame and the rotated frame respectively Case 1: q is the rotation operator w = qv = (q 0 + q)(0 + v) = q 0 0 q v + 0 q + q 0 v + q v = q v + q 0 v + q v For w to be a vector, the real part of the quaternion w must be equal to zero q v = 0 It is clear that we need an operator that takes a vector as an input and returns a vector as the output 10 of 21

18 Constructing the quaternion rotation operator Consider two general quaternions q and r and the vector v. We look at the different products qrv vqr rqv vrq qvr rvq Case 2: qvr or equvialently rvq is the rotation operator w = qvr = (q 0 + q)(0 + v)(r 0 + r) After a lot of algebra, it can shown that the real part of w is given by: If we assume q 0 = r 0, we get: r 0 (q v) q 0 (r v) + (q r) v q 0 (q + r) v + (q r) v It can be clearly seen that the real part is zero if r = q. So, we have r = r 0 + r = q 0 q = q. The quaternion rotation operator is thus given by: L q = qvq or equivalently L q = q vq 11 of 21

19 A few points to remember L q (= q vq) represents a reference frame rotation while L q (= qvq ) represents rotation of the vector If q = cos θ + u sin θ, L q rotates v by 2θ General notation for q is given by: q = cos θ 2 + u sin θ 2 (12) 12 of 21

20 Tracking transformation - matrix approach Rotate about Z axis by α Rotation matrix is given by: cos α sin α 0 A Zα = sin α cos α (13) Rotate about Y 1 axis by β Rotation matrix is given by: cos β 0 sin β A Y1 β = sin β 0 cos β (14) Y 1 Y Z Remote object β X α 13 of 21

21 Tracking transformation - matrix approach Rotate about Z axis by α Rotation matrix is given by: cos α sin α 0 A Zα = sin α cos α (13) Rotate about Y 1 axis by β Rotation matrix is given by: cos β 0 sin β A Y1 β = sin β 0 cos β (14) Remote object β X α Y 1 X 1 Y Z 13 of 21

22 Tracking transformation - matrix approach Rotate about Z axis by α Rotation matrix is given by: cos α sin α 0 A Zα = sin α cos α (13) Rotate about Y 1 axis by β Rotation matrix is given by: cos β 0 sin β A Y1 β = sin β 0 cos β (14) Remote object X 2 β X α Y 1 X 1 Y Z 2 Z 13 of 21

23 Tracking transformation - matrix approach The composite rotation matrix (using eqs. (13) and (14)) is given by: cos β 0 sin β cos α sin α 0 A vθ = sin α cos α 0 sin β 0 cos β cos α cos β sin α cos β sin β = sin α cos α 0 (15) cos α sin β sin α sin β cos β Next step is to find the axis v and the angle θ associated with this composite rotation matrix 14 of 21

24 Finding the composite axis of rotation Let v = x 1ˆ ı + y 1ˆ j + z 1ˆk and since the axis remains unchanged under the composite rotation, we have x 1 y 1 z 1 v = A v v cos α cos β sin α cos β sin β = sin α cos α 0 cos α sin β sin α sin β cos β Rewriting in equation form, we have: x 1 y 1 z 1 (16) x 1 (cos α cos β 1) + y 1 sin α cos β z 1 sin β = 0 (17) x 1 sin α + y 1 (cos α 1) + z 1 0 = 0 (18) x 1 cos α sin β + y 1 sin α sin β + z 1 (cos β 1) = 0 (19) 15 of 21

25 Solving the equations To solve the homogeneous equations, let x 1 = k Eq. (18) yields y 1 = k sin α cos α 1 After substituting x 1 and y 1 in eq. (19) and solving, we get z 1 = sin β cos β 1 After a bit of simplifying, we get ( v = sin α 2 sin β 2, cos α 2 sin β 2, sin α 2 cos β ) 2 We already know from eq. ((11)) [ ] Tr(A) 1 θ = cos 1 2 [ ] cos α cos β + cos α + cos β 1 = cos 1 2 [ = 2 cos 1 cos α 2 cos β ] 2 (20) (21) 16 of 21

26 Summary of matrix approach Composite rotation parameters: v = sin α 2 sin β + cos 2ˆ ı α 2 sin β 2 ˆ j + sin α 2 cos β 2 ˆk (22) [ θ = 2 cos 1 cos α 2 cos β ] (23) 2 17 of 21

27 Tracking transformation - quaternion approach Rotate about Z axis by α Quaternion associated with the rotation operator (L q = q vq) is given by (using eq. (12)): q = cos α 2 + ˆk sin α 2 (24) Remote object X 2 β X α Y 1 X 1 Rotate about Y 1 axis by β Quaternion associated with this rotation is given by: Y Z Z 2 p = cos β 2 + ˆ j sin β 2 (25) 18 of 21

28 Tracking transformation - quaternion approach Quaternion associated with the composite rotation is given by: r = qp ( = cos α 2 + ˆk sin α ) ( cos β ˆ j sin β ) 2 = cos α 2 cos β 2 sin α 2 sin β 2ˆ ı + cos α 2 sin β 2 ˆ j + sin α 2 cos β 2 ˆk (26) We can directly read off the composite rotation parameters from the above as: v = sin α 2 sin β + cos 2ˆ ı α 2 sin β 2 ˆ j + sin α 2 cos β 2 ˆk (27) [ θ = 2 cos 1 cos α 2 cos β ] (28) 2 Eqs. (22), (23) match exactly with eqs. (27) and (28) 19 of 21

29 Quaternions in molecular simulations For MD simulations, it can be shown that choosing a random orientation for molecule reduces to choosing a random unit quaternion on the surface of a sphere For MC simulations, we require small changes to orientations and we do so by a random rotation. This can be achieved by choosing from 1 p(v) = exp( 1 2 v 2 / 2 ) (2π) 3/2 3 Let q old, q new and q rot be the quaternions associated with the old orientation, new orientation and the rotation respectively. We can write q new = q rot q old where q rot = cos v + v sin v 1 Karney J. Mol. Graph. Mod. 25, (2007) 20 of 21

30 Concluding remarks Quaternions are primarily used to perform complex rotation operations quickly and efficiently They provide immediate information about the angle of rotation and the axis Can easily convert from direction cosines, euler angles and rotation matrices to quaternions and vice-versa Few areas of application include a) Aerospace industry b) Computer graphics/gaming industry and c) Molecular simulations 21 of 21