Rotation matrix. Fixed angle and Euler angle. Axis angle. Quaternion. Exponential map

Transcription

1 3D orientation

2 Rotation matrix Fixed angle and Euler angle Axis angle Quaternion Exponential map

3 Joints and rotations Rotational DOFs are widely used in character animation 3 translational DOFs 48 rotational DOFs Each joint can have up to 3 DOFs 1 DOF: knee 2 DOF: wrist 3 DOF: arm

4 Representation of orientation Homogeneous coordinates (review) 4X4 matrix used to represent translation, scaling, and rotation a point in the space is represented as p = Treat all transformations the same so that they can be easily combined x y z 1

5 Translation x + t x y + t y z + t z 1 = t x t y t z x y z 1 new point translation matrix old point

6 Scaling s x x s y y s z z 1 = s x s y s z x y z 1 new point scaling matrix old point

7 Rotation x y z 1 = cosθ sin θ 0 0 sin θ cos θ x y z 1 X axis x y z 1 = cos θ 0 sin θ sin θ 0 cosθ x y z 1 Y axis x y z 1 = cos θ sin θ 0 0 sin θ cos θ x y z 1 Z axis

8 Composite transformations θ 1 x 0 y 0 z 0 θ 0 φ 0 σ 0 φ 1 σ 1 θ 3 φ 3 h 0 θ 2 h 1 h 2 p h 3 (h 3, 0, 0) A series of transformations on an object can be applied as a series of matrix multiplications p : position in the global coordinate x : position in the local coordinate p = T(x 0,y 0,z 0 )R(θ 0 )R(φ 0 )R(σ 0 )T(0,h 0, 0)R(θ 1 )R(φ 1 )R(σ 1 )T(0,h 1, 0)R(θ 2 )T(0,h 2, 0)R(θ 3 )R(φ 3 )x

9 Interpolation In order to move things, we need both translation and rotation Interpolation the translation is easy, but what about rotations?

10 Interpolation of orientation How about interpolating each entry of the rotation matrix? The interpolated matrix might no longer be orthonormal, leading to nonsense for the inbetween rotations

11 Interpolation of orientation Example: interpolate linearly from a positive 90 degree rotation about y axis to a negative 90 degree rotation about y Linearly interpolate each component and halfway between, you get this

12 Properties of rotation matrix Easily composed? Yes Interpolate? No

13 Rotation matrix Fixed angle and Euler angle Axis angle Quaternion Exponential map

14 Fixed angle Angles used to rotate about fixed axes Orientations are specified by a set of 3 ordered parameters that represent 3 ordered rotations about fixed axes Many possible orderings

15 Euler angle Same as fixed angles, except now the axes move with the object An Euler angle is a rotation about a single Cartesian axis Create multi-dof rotations by concatenating Euler angles evaluate each axis independently in a set order

16 Euler angle vs. fixed angle R z (90)R y (60)R x (30) = E x (30)E y (60)E z (90) Euler angle rotations about moving axes written in reverse order are the same as the fixed axis rotations Z Y X

17 Properties of Euler angle Easily composed? No Interpolate? Sometimes How about joint limit? Easy What seems to be the problem? Gimbal lock

18 Gimbal Lock A Gimbal is a hardware implementation of Euler angles used for mounting gyroscopes or expensive globes Gimbal lock is a basic problem with representing 3D rotation using Euler angles or fixed angles

19 Gimbal lock When two rotational axis of an object pointing in the same direction, the rotation ends up losing one degree of freedom

20 Rotation matrix Fixed angle and Euler angle Axis angle Quaternion Exponential map

21 Axis angle Represent orientation as a vector and a scalar vector is the axis to rotate about scalar is the angle to rotate by y z x

22 Properties of axis angle Can avoid Gimbal lock. Why? It does 3D orientation in one step Can interpolate the vector and the scalar separately. How?

23 Axis angle interpolation θ k =(1 k)θ 1 + kθ 2 A 1 θ 1 z y A 2 θ2 x B = A 1 A 2 ( ) φ = cos 1 A1 A 2 A 1 A 2 A k = R B (kφ)a 1

24 Properties of axis angle Easily composed? No, must convert back to matrix form Interpolate? Yes Joint limit? Yes Avoid Gimbal lock? Yes

25 Rotation matrix Fixed angle and Euler angle Axis angle Quaternion Exponential map

26 Quaternion θ 1 θ 2 (θ 1,φ 1 ) (θ 2,φ 2 ) 1-angle rotation can be represented by a unit circle 2-angle rotation can be represented by a unit sphere What about 3-angle rotation?

27 Quaternion 4 tuple of real numbers: w, x, y, z q = w x y z = [ w v ] scalar vector Same information as axis angles but in a different form θ r q = [ cos (θ/2) sin (θ/2)r ]

28 Quaternion math Unit quaternion q =1 x 2 + y 2 + z 2 + w 2 =1 Multiplication [ w1 v 1 ][ w2 v 2 ] = [ w 1 w 2 v 1 v 2 w 1 v 2 + w 2 v 1 + v 1 v 2 ] q 1 q 2 q 2 q 1 q 1 (q 2 q 3 )=(q 1 q 2 )q 3

29 Quaternion math Conjugate q = [ w v ] = [ w v ] (q ) = q (q 1 q 2 ) = q 2q 1 Inverse q 1 = q q qq 1 = identity quaternion

30 Quaternion Rotation z θ y r p x q p = [ 0 p ] q = [ cos (θ/2) sin (θ/2)r ] If q is a unit quaternion and then qq p q 1 results in p rotating about r by θ proof: see Quaternions by Shoemaker

31 Quaternion Rotation qq p q 1 = = = [ w v [ w v [ ][ 0 p ][ ][ w v ] p v wp p v ] wp v v wp + v p v =0 w(wp p v)+(p v)v + v (wp p v) ] [ w1 v 1 ][ w2 v 2 ] = [ w 1 w 2 v 1 v 2 w 1 v 2 + w 2 v 1 + v 1 v 2 ]

32 Quaternion composition If q 1 and q 2 are unit quaternion the combined rotation of first rotating by then by is equivalent to q 2 q 1 and q 3 = q 2 q 1

33 Matrix form q = w x y z R(q) = 1 2y 2 2z 2 2xy +2wz 2xz 2wy 0 2xy 2wz 1 2x 2 2z 2 2yz +2wx 0 2xz +2wy 2yz 2wx 1 2x 2 2y

34 Quaternion interpolation θ 1 θ 2 (θ 1,φ 1 ) (θ 2,φ 2 ) 1-angle rotation can be represented by a unit circle 2-angle rotation can be represented by a unit sphere Interpolation means moving on n-d sphere

35 Quaternion interpolation Moving between two points on the 4D unit sphere a unit quaternion at each step - another point on the 4D unit sphere move with constant angular velocity along the great circle between the two points on the 4D unit sphere

36 Quaternion interpolation Direct linear interpolation does not work Linearly interpolated intermediate points are not uniformly spaced when projected onto the circle Spherical linear interpolation (SLERP) θ slerp(q 1, q 2,u)=q 1 sin((1 u)θ) sin θ + q 2 sin(uθ) sin θ Normalize to regain unit quaternion

37 Quaternion constraints Cone constraint q = w x y z θ 1 cos θ 2 = y 2 + z 2 Twist constraint θ tan (θ/2) = q axis w

38 Properties of quaternion Easily composed? Interpolate? Joint limit? Avoid Gimbal lock? So what s bad about Quaternion?

39 Rotation matrix Fixed angle and Euler angle Axis angle Quaternion Exponential map

40 Exponential map Represent orientation as a vector direction of the vector is the axis to rotate about magnitude of the vector is the angle to rotate by Zero vector represents the identity rotation

41 Properties of exponential map No need to re-normalize the parameters Fewer DOFs Good interpolation behavior Singularities exist but can be avoided

42 Choose a representation Choose the best representation for the task input: Euler angles joint limits: Euler angles, quaternion (harder) interpolation: compositing: axis angle, quaternion or exponential map quaternions or orientation matrix rendering: orientation matrix ( quaternion can be represented as matrix as well)

43 Summary What is a Gimbal lock? What representations are subject to Gimbal lock? How does the interpolation work in each type of rotations?

44 What s next?

45 Physics! Ordinary differential equations Numeric solutions Read: Quaternions by Ken Shoemake