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Introduction The Latest Springy Fashions Problems to Avoid and Ignore

The Latest Springy Fashions

The mass and spring dynamics simulation I developed in a previous column ("Collision Response: Bouncy, Trouncy, Fun") proved effective for simulating soft body objects in real time. I thought it should be possible to use these techniques to create a cloth simulation. In fact, several of the commercial cloth animation systems for 3D animation programs such as 3D Studio Max, Softimage, and Maya use similar techniques. So how do I go about creating a piece of cloth?

I am going to be using the same spring force formulas for the cloth simulation as the ones I used in the March  column. If you are unfamiliar with the dynamic forces generated by springs, you should go back and read the March column or at least take a look at the March source code on the Game Developer web site (http://www.gdmag.com).

Figure 1A. A simple cloth grid.

I start by creating a rectangular grid, and then connect each point to neighboring points with springs, as you can see in Figure 1A. These springs define the rough structure of the cloth and so I refer to them as structural springs. The resulting cloth patch looks pretty good and requires few springs. However, once I run the simulation, problems appear immediately as shown in Figure 1B.

 

Figure 1B. Stretched cloth grid.

The simple spring connections are not enough to force the grid to hold its shape. Much like the box in the March column, there are simply no springs to maintain the shape. If I held on to only one point, the entire surface would collapse into a single line creating a rope. Not exactly what I wanted, but it points out something I want to address a bit later.

I really want to keep the model from shearing too much. That is, I want the space between diagonal elements of the model preserved. So, I just add a few more springs to the grid along the diagonals creating a group of shear springs, as you can see in Figure 2A. Run this new structure through the simulation and the results are much better, as you see in Figure 2B.

Figure 2a. Added shear springs.

 

Figure 2B. A better cloth model.

This new form of cloth works pretty well hanging from hooks on the wall. However, if you drop the cloth on the floor, it wads up into a big mass of springy spaghetti. The reason for this failure is that the model is still incomplete. If you look at the structure in Figure 2A, you may see that there is nothing to keep the model from folding along the edges of the structural springs, much as you fold a handkerchief. The fibers that comprise actual cloth run the length of the fabric and generally resist folding and bending. In order to simulate this effect adequately, I need to do a little more work.

My research uncovered two methods for dealing with this problem. The first minimized the bend between two adjacent cells by using the dot product to determine the angle of bend. The second method simply added an extra set of springs called flexion or bend springs to apply the bend force. I created the bend springs by stretching a spring across two cells alongside the structural springs. These springs end up connecting every other cell in the cloth mesh.

Figure 3. The interconnection of cloth springs.

I prefer the second method because it works within the existing spring system without the need for a new method for calculating forces. I also get the benefit of having only one calculation to optimize later. For other applications, it’s possible that the angle minimization method may work out better.

I now have a sufficient spring network to simulate a variety of different types of cloth. You can see how all the springs are connected in Figure 3.

Stretch Without Tearing

These three types of springs make it possible to simulate a variety of different cloth types. By varying the stiffness of the springs, it’s possible to simulate anything from stiff cardstock to stretchy nylon. For example, spandex would have very flexible structural and shear springs to allow strong stretching capability. Paper, on the other hand, is very resistant to shearing and stretching, so its springs would be very stiff. When considering how much a material will bend, a surface such as cardboard should have the very stiff bend springs to make it resistant to folding. I find experimenting with different values for spring stiffness the only real way to find adequate surface properties.

Stiff springs can make a numerical simulation unstable. To combat this, it’s important to use a good numerical integrator. The midpoint method and Runge-Kutta integrators developed last month seem to do the trick nicely.

Making It Move

Figure 4. Collision boxes allow for the draping effect on this tablecloth.

I already have a simulator from the March column that is capable of handling a cloth patch. I can even apply gravity to it and lock the position of individual vertices. That’s pretty interesting, but it needs some improvement to come alive. In March, I also discussed the use of planes for collision. With this same method, I can create collision boxes that enable me to simulate a tablecloth draped over a table, as you see in Figure 4.

This model is interesting and realistic looking but not terribly animated. In fact, in this case it’s probably better to freeze the simulation and avoid the constant recalculation. Unless, of course, the wind kicks up or someone pulls on the corner.

Figure 5. Collision with a sphere.

For characters, a moving box is not the most realistic way to displace the cloth. Moving bounding spheres allow much more pleasing character animation. Fortunately, this is easy to add to the simulation. Determining whether a point is inside a sphere is very easy. If the distance from the point to the center of the sphere is less than the radius of the sphere, the point is on the inside. If a point in the cloth is found inside a sphere, I have a penetrating collision. Just like handling collisions in the March simulator, I need to back up the simulation time to find the actual point of contact. In a sphere, contact takes place when the distance of the point to the sphere’s center is equal to the radius of the sphere. Now that the contact point has been established, I need to resolve the collision. The collision normal, N, between a point and a sphere is the vector between the point of contact and the center of the sphere. You can see this in Figure 5. Fortunately for me, the rest of the collision response is handled just like the collision with a plane. This means my existing collision response code works great.

Figure 6. Cloth hanging from a pair of invisible hips.

I can now add collision spheres to my simulation. The cloth slides realistically off the spheres. You can see how the simulation looks with two collision spheres in Figure 6. By animating the spheres along with the 3D model, I can get a nice animated hip sway and other alluring effects. The motion of the cloth continues after the animation stops, creating entertaining effects that are diffiicult to achieve with traditional animation techniques.

 

 

 

 

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Problems to Avoid and Ignore