3D Modelling: Animation Fundamentals & Unit Quaternions

Transcription

1 3D Modelling: Animation Fundamentals & Unit Quaternions CITS3003 Graphics & Animation Thanks to both Richard McKenna and Marco Gillies for permission to use their slides as a base.

2 Static objects are boring Look at your project part 1 scene objects they just sit there we can move them to other places and alter the rotation and colour, but they re still pretty boring What about more complicated & interesting shapes? living things vehicles etc. These things are all made up of separate moving parts that are still part of the whole object where I go, my arms go with me 2 Jack Skellington, from The Nightmare Before Christmas

3 Example: Game Player Animation States Think about a First Person Game What game states does the player have? running forward running backwards running sideways jumping falling landing dying (in all sorts of ways) 3 etc. Jack Skellington, from The Nightmare Before Christmas

4 Typical Role of Artist Create a mesh for a game shape/character/vehicle geometry texturing materials Create animation sequences for different game states for that entity NOTE: exporting data from modelling/animation tools so that it can be imported into another application can be tricky. 4

5 Keyframe (Cell) Animation In traditional hand-drawn animation: the senior key artist would draw the keyframes, and the junior artist would fill the in-between frames. We can use a similar approach for computer animation For a 30-fps animation, less than 30 keyframes need to be defined per second: 1. The data (e.g., joint angles of an articulated figure) are assigned to the keyframes 2. In-between frames are interpolated 10-5

6 Again, the positions between frames? The positions between frames must be interpolated. Linear interpolation is commonly used. Assume a time from 0.0 to 1.0 from one keyframe to the next pp ii = pp 1 + tt ii (pp 2 pp 1 ) where pp and pp 1 2 are positions in keyframes 1 and 2; ii is the index of the frame (between the two keyframes) to be interpolated; tt ii =

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8 A Simple Animation Example: A Tank Gun Turret should be able to rotate Gun Barrel should be able to move up and down The wheels should be able to spin Note: keyframes do not need to be equi-distant apart. Keyframe 1: set the initial position of the gun turret Keyframe 2: set the new desired position of the gun turret Keyframe 3: set the new desired position of the gun barrel 8

9 What would be the Tank Hierarchy? Tank hull this is the central parent section Wheels these are children of the hull and are siblings to each other Gun Turret this is another child of the hull Gun Barrel this is the only child of the Gun Turret Thus, where the hull goes, the wheels go when the hull rotates, the wheels rotate Note that the hull, wheels, etc., are represented as 3D meshes (i.e., triangles), so when the hull moves, it is actually the vertices of the triangles being transformed. 9

10 Problems With Interpolating Euler Angles The Euler angles (pitch, yaw, and roll) are often used (e.g. in blender) to represent the rotation angles of a body limb about a body joint, e.g., In keyframe 1 (frame number 11): we have αα 1, ββ 1, and γγ 1 representing the pitch, yaw, and roll angles of a rotation. In keyframe 2 (frame number 21): we have αα 2, ββ 2, and γγ 2 representing the pitch, yaw, and roll angles of another rotation. We need to compute these angles for frames 12 to 20. Unfortunately, the Euler angles are problematic when interpolating: Unexpected rotations would result when we combine the three interpolated angles, e.g. consider interpolating between αα 1 = 170 and αα 2 = 170. Gimbal lock the second transform may align the first and third, leading to 2 degrees of freedom rather than 3 10

11 Gimbal Lock Gimbal lock refers to the situation where a rotation accidentally brings two local coordinate axes align and thus losing 1 degree of freedom (i.e., we are left with two affective rotation axes rather than three). This is a serious problem as using the Euler angles to represent rotations we are not able to rotate about one of the 3 principal axes (xx, yy, and zz) when gambal lock occurs. Some useful youtube demos and explanation about gimbal lock:

12 Can We Avoid Gimbal Lock? Can we avoid gimbal lock in interpolating rotations of keyframes if we want to remain using the Euler angles? Consider the three Euler angles: pitch (θθ xx ), yaw (θθ yy ), and roll (θθ zz ) angles. We can use any of the 6 multiplication orders: 1. RR xx θθ xx RR yy θθ yy RR zz (θθ zz ) 2. RR xx θθ xx RR zz θθ zz RR yy (θθ yy ) 3. RR yy θθ yy RR xx θθ xx RR zz (θθ zz ) 4. RR yy θθ yy RR zz θθ zz RR xx (θθ xx ) 5. RR zz θθ zz RR xx θθ xx RR yy (θθ yy ) 6. RR zz θθ zz RR yy θθ yy RR xx (θθ xx ) Note: as multiplication of matrices is not commutative, the resultant rotation matrix is different for each order. You will need to work out the appropriate Euler angles for each order, so that the resultant rotation matrix is the desired one. If using multiplication order #6 leads to a gimbal lock problem, say, then we can use another multiplication order. Note that this does not eliminate gimbal lock; we only avoid the problem for a particular 12 interpolation.

13 Quaternions An alternative solution is to use unit quaternions. Quaternions are an extension of complex numbers. They are often written as aa + bbbb + cccc + dddd, where aa, bb, cc, dd R, and ii, jj, kk satisfy ii 2 = jj 2 = kk 2 = iiiiii = 1. An alternative representation (and more suitable for representing rotations) is consider quaternions as 4D-vectors qq = (aa, bb, cc, dd). For compactness, we often write this as qq = (aa, vv), where aa R and vv R 3. When aa = 0, we refer to the quaternion as a pure quaternion; when qq = 1, we refer to it as a unit quaternion. Rotations can be represented using unit quaternions. 13

14 Quaternion Advantages Interpolation is easy between 2 quaternions. Smoother animation can be achieved. Compared to matrices, quaternions take up less storage space. Quaternions can be easily converted to matrices for rendering. Quaternions do not suffer from gimbal lock. 14

15 Unit Quaternions are related to Angle-Axis Rotations We use unit quaternions to represent 3D rotations, i.e., if we use qq = (aa, vv) to represent a rotation then we normalize qq so that qq = 1. Any 3D rotation about the origin can be described by defining A rotation axis ww = (ww 1, ww 2, ww 3 ) (assuming that ww = 1) in the 3D space A rotation angle θθ about that axis The corresponding unit quaternion is cos θθ 2, ww sin θθ 2 15

16 Unit Quaternions A Few Examples Rotation about the xx-axis an angle αα: the unit quaternion is cos αα 2, sin αα 2, 0, 0 Rotation about the yy-axis an angle ββ: the unit quaternion is cos ββ 2, 0, sin ββ 2, 0 Rotation about the zz-axis an angle γγ: the unit quaternion is cos γγ 2, 0, 0, sin γγ 2 We know that the inverse of a rotation matrix RR is RR TT. Let qq = (aa, vv) be the quaternion representing RR, then qq 1 = (aa, vv) is the inverse of qq representing RR TT. The quaternion (1, 0, 0, 0) represents the identity matrix (i.e., zero rotation angle). 16

17 Quaternion Multiplication Multiplying two rotation matrices together is equivalent to multiplying two quaternions. The multiplication of two quaternions is defined as follows: Let Then qq 1 = ss 1, vv 1 qq 2 = ss 2, vv 22 qq 1 qq 2 = (ss 1 ss 2 vv 1 vv 2, ss 1 vv 2 + ss 2 vv 1 + vv 1 vv 2 ) Here is cross product, and is dot product. Same as matrix multiplication, quaternion multiplication 17 is not commutative.

18 Quaternion Multiplication (cont.) An easy example that we can verify: Let rotation RR 1 be the rotation rotation x (αα) and let RR 2 be rotation x (ββ). Then we know the resultant rotation RR should be rotation x (αα + ββ). The corresponding unit quaternions are: qq 1 = (cos αα, sin αα, 0, 0) and qq = (cos ββ, sin ββ, 0, 0). 2 2 Applying the multiplication formula, we get cos αα 2 cos ββ 2 sin αα 2 sin(ββ 2) qq = cos αα 2 sin ββ 2 + cos ββ 2 sin(αα 2) 0 0 Recall that qq 1 qq 2 = (ss 1 ss 2 vv 1 vv 2, ss 1 vv 2 + ss 2 vv 1 + vv 1 vv 2 ). Here we have ss 1 = cos(αα 2), ss 2 = cos(ββ 2), vv 1 = (sin αα 2, 0, 0), and vv 2 = (sin ββ 2, 0, 0). 18

19 Quaternion Multiplication (cont.) Using the rules cos AA + BB = cos AA cos BB sin AA sin(bb) and sin AA + BB = sin AA cos BB + cos AA sin (BB) we get the resultant quaternion cos( αα+ββ 2 ) qq = sin αα+ββ which represents the rotation about the x-axis an angle αα + ββ. Recall that qq 1 qq 2 = (ss 1 ss 2 vv 1 vv 2, ss 1 vv 2 + ss 2 vv 1 + vv 1 vv 2 ). Here we have ss 1 = cos(αα 2), ss 2 = cos(ββ 2), vv 1 = (sin αα 2, 0, 0), and vv 2 = (sin ββ 2, 0, 0). 19

20 Rotating Points and Vectors using Quaternions Let pp = (xx, yy, zz) and vv = (uu, vv, ww) be a point and a vector in 3D. Let qq = (aa, ww) be the unit quaternion representing the 3D rotation matrix RR. Suppose that we want to transform pp and vv by RR. i.e., we want to compute pp = RRRR and vv = RRRR. How do we achieve the same result using the unit quaternion qq? We need to rewrite both pp and qq as 4D entities by letting pp = (0, xx, yy, zz) and vv = (0, uu, vv, ww). Then pp = qq pp qq 1 vv = qq vv qq 1 where qq 1 denotes the inverse of qq. As unit quaternions cannot be used for translation, it doesn t matter whether we have a 0 or a 1 here. Both the outputs ppp and vvv are 4D entities. Their 1 st components would be 0. To convert them back to the 3D space, we can simply drop the 1 st component. Unit quaternions cannot be used for scaling and translation. 20

21 Interpolating Quaternions 2 ways of interpolating quaternions LERP (Linear interpolation) LERP(qq 0, qq 1, tt) = (1 tt)qq 0 + ttqq 1 The interpolated quaternion for a given t value. qq 1 LERP θθ SLERP qq 0 SLERP (spherical linear interpolation) SLERP qq 0, qq 1, tt = sin (1 tt)θθ qq 0 sin (θθ) + sin (ttθθ)qq 1 sin (θθ) where qq 0 and qq 1 are the quaternions that describe the initial and final orientations (= rotation), parameter tt = 0 1, and θθ is the angle between qq 0 and qq 1, i.e., θθ = arccos qq 0 qq 1 21

22 Interpolating Quaternions (cont.) SLERP qq 0, qq 1, tt = sin (1 tt)θθ sin (θθ) qq 0 + sin (ttθθ) sin (θθ) qq 1 Note that SLERP is usually implemented slightly differently in practice. Also, normalization would be required since the two weights do not sum to 1. What we aim to achieve is the shortest trajectory (the geodesic) that brings the initial orientation qq 0 of the object to its final orientation qq 1. Mathematically, we find qq such that qq 1 = qq qq 0, i.e., 1 qq = qq 1 qq 0 then at each tt 0,1, the interpolated quaternion is computed as qq tt qq weights

23 Interpolating Quaternions (cont.) One last thing to note: The unit quaternion is not a unique representation for rotations. E.g, both qq = (aa, vv) and qq = ( aa, vv) represent the same rotation. If one corresponds to a 160 rotation then the other corresponds to a 200 in the other direction. To avoid interpolating along a longer path, we should change the sign of the quaternion so that its 1 st component (representing cos θθ/2 ) is always positive. This will ensure that the rotation angle θθ is always less than

24 The easiest way to convert a unit quaternion qq = (aa, vv) to a 3 3 rotation matrix RR is using the Rodriques rotation formula: where Converting Unit Quaternions to Rotation Matrices RR = II + sin θθ skew vv + 1 cos θθ vv vv TT RR TT = II sin θθ skew vv + 1 cos θθ vv vv TT II denotes the 3 3 identity matrix; θθ can be obtained from the 1 st component of qq as follows: θθ = 2 arccos aa ; skew vv denotes the skew-symmetric matrix composed from the elements of vv = (vv 1, vv 2, vv 3 ): 0 vv 3 vv 2 skew vv = vv 3 0 vv 1 vv 2 vv

25 Quaternions in OpenGL? OpenGL doesn t support them directly, but implementing quaternions is not difficult. So what do we do? 1. do all the rotation interpolation between frames using unit quaternions. 2. convert the interpolated results to matrices and combine them with the necessary translation and scaling matrices. 3. pass the resultant matrices to the vertex shader. Can also use dual quaternions instead: [ Dual quaternions have recently been used for better blending by representing combined rotations and translations. See: ] 25

26 Summary In summary, to generate smooth animation, we do the following: Define the pose (orientation/rotation and position) of each object in keyframes. Option 1 (preferred): Try to use quaternions to represent rotations to avoid gimbal lock. Interpolate quaternions to get the rotations for the in-between frames. Get the resultant rotation matrices for these frames. Option 2: Try to avoid gimbal lock by change the order of applying the roll, pitch, and yaw in the rotation. Will need different orders for different parts of the animation. Combine the rotation with scaling and translation etc to get the resultant transformation matrices for the vertex shader. 26

27 References Useful references on the web: Wikipedia: Youtube: Quaternion and quaternion interpolation : Comparing LERP and SLERP: 27