Fundamentals of Computer Animation

Transcription

1 Fundamentals of Computer Animation Flexible Objects (1) page 1

2 Flexible Objects Elastic and inelastic behavior, viscoelasticity, plasticity, fracture Elastically Deformable Models Terzopoulos et al SIGGRAPH 87 page 2

3 Modeling Inelastic Deformation: Viscoelasticity, Plasticity, Fracture Terzopoulos and Fleiseher SIGGRAPH 88

4 Graphical Modeling and Animation of Brittle Fracture O Brien and Hodgins SIGGRAPH 99 Simulation of Object and Human Skin Deformations in a Grasping Task Gourred et al SIGGRAPH 89 page 4

5 Graphical Modeling and Animation of Ductile Fracture O Brien et al SIGGRAPH 02 page 5

6 Spring-Mass Systems Model objects as systems of springs and masses The springs exert forces, and you control them by changing their rest length A reasonable, but simple, physical model for muscles Advantage: Good looking motion when it works Disadvantage: Expensive and hard to control page 6

7 Springs (Hooke s law) Spring s rest length: exerts zero force F spring = k spring (x x rest ) x x rest F x F F F page 7

8 SPRING-MASS SYSTEMS The simplest, most common approach Straightforward strategy: Point Mass Spring (rest length = edge length) External Forces (collisions, gravity, wind, ) page 8

9 Spring Mass System V1 External Force V2 V3 spring force F i,j = -F j,i = k(dist i,j (t) len i,j )v i,j dist i,j is the distance at time t and len i,j is the rest length v i,j is a vector in the direction v i to v j. page 9

10 Damping If each time step assumes constant acceleration so the simulation can gain energy and explode. Use a damper which imparts a force against velocity. Damping force i = -k d v i (t) As spring changes length faster and faster damper helps to control change. Damping force is proportional to relative velocity of end points in a mesh. page 10

11 Damping Calm down spring oscillations F damping = k damping v F = k ( x x ) spring rest k damping v page 11

12 Spring Mesh Edges => springs Internal springs to stabilize shape page 12

13 Spring mass fish Due to Xiaoyuan Tu, page 13

14 Spring mass fish page 14

15 Angular Springs An angular spring imparts a restoring torque ie it resists deviation to the rest angle. θ τ = k spring ( θ ( t) θ ) k & θ ( t) rest damping page 15

16 Virtual Springs To increase control. (constraints) Forces can be introduced that do not model physical elements E.g. penalty method: Virtual spring with zero rest length can make one object lie on another or at some given distance apart (rest length>0). Proportional Derivative Controller Keep control variable and derivative within desired values. e.g. To maintain joint angle and joint velocity close to desired values introduce torque: τ = k spring ( θ ( t) θ ) k ( & θ ( t) & θdesired) desired damping page 16

17 Strings A whole line of points attached together with springs Simple to model, great for making realistic straps of bullets for chain guns, tails on animals, bungie ropes. The springs have a normal length of, say, one unit. If the adjacent points move further than one unit of length apart, they experience a force towards each other proportional to the extension of the spring that connects them. Likewise, if they move closer than one unit apart, they experience a force pushing them apart. page 17

18 Strings Two ways to model the force on the points With mass If you are creating animations Without mass If you are just trying to find the optimum shape of a string hanging over a certain object Forces between Two Springs page 18

19 Strings without Mass Forces affect the position of the point c i a v ac f = v ab vab + l ab + v ac vac + l ac v ab s 1 b f = f g Normal length β s s + 2 = 1 f gravity Small amount (0.01 or so): makes the string move slowly page 19

20 Strings with Mass c Forces affect the velocity of the point i a v ac f = v ab vab + l ab + v ac vac + l ac s1 b v ab f v = f g β = v ω + f s s + 2 = 1 v If you make a string like this, you will notice that it is extremely flexible. To make it stiffer, you can compare each point with its 4 or even 6 closest neighbors, instead of 2. Damping (between about 0.95 and 0.99), is the energy loss from the string. If you set it to 1, then the string will never stop swinging around, and setting it to more than 1 will make the string increase its swing by itself and eventually fly off the screen. page 20

21 Cloth Simply a whole load of interwoven strings! We need to add an extra dimension to our string routine. Imagine a cloth to be a sheet of points all connected together by springs. If two points get pulled further apart, then they experience a force pulling them together and vice versa. This very simple model of a cloth is reasonably accurate! Stanford Cloth Demo page 21

22 Cloth Behavior If you compare each point with its 4 nearest neighbors a fisherman's net. If you compare each point with its 8 nearest neighbors a very flexible cloth If you compare each point with its 24 nearest neighbors a more realistic, stiffer cloth, though it's much slower to compute page 22

23 Massless Cloths Every point on the cloth moves at a rate proportional to the sum of the forces acting on it from the neighboring points. Create a 2-dimensional array of co-ordinates to hold the x, y and z positions of the cloth in space. Initialize the values of cloth(p,q) to (p,q,0). We will need two of these arrays. One to hold the current state of the cloth, and the other to hold the new cloth that is being calculated. When we have finished calculating the cloth, copy all the values from our second array back to the first. page 23

24 cloth1 (0 to 31, 0 to 31) cloth2 (0 to 31, 0 to 31) Variables: VECTOR: MovementVector VECTOR: SpringVector VECTOR: ForceVector VECTOR: Gravity (initialised to (0, 0, g) where g is gravity, 0.02 is a good number) REAL: Length REAL: ForceScaler REAL: NormalLength page 24

25 For every point (p,q) on the cloth: MovementVector = Gravity For each of the 24 neighboring points: SpringVector = (position in space of neighbour) - (position in space of point (p,q)) Length = length of SpringVector NormalLength = The length SpringVector would be if the cloth were unstretched ForceScaler = (Length - NormalLength) / NormalLength SpringVector = SpringVector * (1/Length) ForceVector = SpringVector * ForceScaler ForceVector = ForceVector * SmallAmount add ForceVector to MovementVector end of loop Add MovementVector to cloth1(p,q) and store it in cloth2(p,q) make sure this point does not move inside an object end of loop Copy all the values in cloth2 to cloth1 keep doing all this forever

26 Cloth Interacting with Objects We will need some objects for the cloth to interact with. The simplest is a floor. Check each point on the cloth to see if it is below the floor, and if it is, then move it to the surface. It is quite easy to make a sphere for the cloth to fall over! Check each point to see if it is inside the sphere. If it is, then move it to the nearest point on the surface of the sphere. page 26

27 Cloth with Sphere REAL: Distance Distance = distance from the point(p,q) to the center of the sphere if Distance < (radius of sphere) then: end if ForceVector = (position of point in space) - (center of sphere) ForceVector = Forcevector / Distance * radius point(p,q) = (center of sphere) + ForceVector page 27

28 Adding Wind Adding wind to the cloth allows us to simulate the fluttering of flags and other cloth+wind kind of situations. This model is not totally accurate. The wind affects the cloth, but the cloth does not affect the wind, to do this would require a vast amount of fluid dynamic calculation. However, it produces reasonable looking fluttering effects. For this we will need to be modeling cloth with mass. page 28

29 Adding Wind First the cloth must be broken down into triangles. This is easy to do, since the cloth is already described as an array of points. The effect of the wind on the cloth is calculated on each of these triangles individually. At each point of the cloth, the sum of the effect of the wind on the surrounding triangles is calculated. page 29

30 Adding Wind The force acting on a triangle due to air molecules bouncing off it will always be in the direction of the normal vector of that triangle. The normal vector for each triangle will have to be calculated every frame because it will be constantly changing. page 30

31 Adding Wind The force will be proportional to the surface area of the triangle, the angle at which the wind hits the triangle, and the speed of the wind. When we use the Cross Product to calculate the normal vector of the triangle, the length of that vector is proportional to the area of the triangle, which makes things a little simpler. page 31

32 VECTOR: force VECTOR: normal VECTOR: wind set force vector to (0,0,0) on all points on cloth loop through all triangles force = unitvector(normal) * dotproduct(normal, wind) add force to all points making up this triangle end of loop loop through all points on cloth add gravity to force add force to velocity end of loop -- rest of cloth routine --