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Monday, December 20, 2010

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Advanced Physics

Advanced Character Physics

Abstract
This paper explains the basic elements of an approach to physically-based modeling which is well suited for interactive use. It is simple, fast, and quite stable, and in its basic version the method does not require knowledge of advanced mathematical subjects (although it is based on a solid mathematical foundation). It allows for simulation of both cloth; soft and rigid bodies; and even articulated or constrained bodies using both forward and inverse kinematics.

The algorithms were developed for IO Interactive’s game Hitman: Codename 47. There, among other things, the physics system was responsible for the movement of cloth, plants, rigid bodies, and for making dead human bodies fall in unique ways depending on where they were hit, fully interacting with the environment (resulting in the press oxymoron “lifelike death animations”).

The article also deals with subtleties like penetration test optimization and friction handling.

1 Introduction
The use of physically-based modeling to produce nice-looking animation has been considered for some time and many of the existing techniques are fairly sophisticated. Different approaches have been proposed in the literature [Baraff, Mirtich, Witkin, and others] and much effort has been put into the construction of algorithms that are accurate and reliable. Actually, precise simulation methods for physics and dynamics have been known for quite some time from engineering. However, for games and interactive use, accuracy is really not the primary concern (although it’s certainly nice to have) – rather, here the important goals are believability (the programmer can cheat as much as he wants if the player still feels immersed) and speed of execution (only a certain time per frame will be allocated to the physics engine). In the case of physics simulation, the word believability also covers stability; a method is no good if objects seem to drift through obstacles or vibrate when they should be lying still, or if cloth particles tend to “blow up”.

The methods demonstrated in this paper were created in an attempt to reach these goals. The algorithms were developed and implemented by the author for use in IO Interactive’s computer game Hitman: Codename 47, and have all been integrated in IO’s in-house game engine Glacier. The methods proved to be quite simple to implement (compared to other schemes at least) and have high performance.

The algorithm is iterative such that, from a certain point, it can be stopped at any time. This gives us a very useful time/accuracy trade-off: If a small source of inaccuracy is accepted, the code can be allowed to run faster; this error margin can even be adjusted adaptively at run-time. In some cases, the method is as much as an order of magnitude faster than other existing methods. It also handles both collision and resting contact in the same framework and nicely copes with stacked boxes and other situations that stress a physics engine.

In overview, the success of the method comes from the right combination of several techniques that all benefit from each other:
• A so-called Verlet integration scheme.
• Handling collisions and penetrations by projection.
• A simple constraint solver using relaxation.
• A nice square root approximation that gives a solid speed-up.
• Modeling rigid bodies as particles with constraints.
• An optimized collision engine with the ability to calculate penetration depths.

Each of the above subjects will be explained shortly. In writing this document, the author has tried to make it accessible to the widest possible audience without losing vital information necessary for implementation. This means that technical mathematical explanations and notions are kept to a minimum if not crucial to understanding the subject. The goal is demonstrating the possibility of implementing quite advanced and stable physics simulations without dealing with loads of mathematical intricacies.

The content is organized as follows. First, in Section 2, a “velocity-less” representation of a particle system will be described. It has several advantages, stability most notably and the fact that constraints are simple to implement. Section 3 describes how collision handling takes place. Then, in Section 4, the particle system is extended with constraints allowing us to model cloth. Section 5 explains how to set up a suitably constrained particle system in order to emulate a rigid body. Next, in Section 6, it is demonstrated how to further extend the system to allow articulated bodies (that is, systems of interconnected rigid bodies with angular and other constraints). Section 7 contains various notes and shares some experience on implementing friction
etc. Finally, in Section 8 a brief conclusion.

In the following, bold typeface indicates vectors. Vector components are indexed by using subscript, i.e., x=(x1, x2, x3).

2 Verlet integration
The heart of the simulation is a particle system. Typically, in implementations of particle systems, each particle has two main variables: Its position x and its velocity v. Then in the time-stepping loop, the new position x’ and velocity v’ are often computed by applying the rules



where t is the time step, and a is the acceleration computed using Newton’s law f=ma (where f is the accumulated force acting on the particle). This is simple Euler integration.

Here, however, we choose a velocity-less representation and another integration scheme: Instead of storing each particle’s position and velocity, we store its current position x and its previous position x*. Keeping the time step fixed, the update rule (or integration step) is then



This is called Verlet integration (see [Verlet]) and is used intensely when simulating molecular dynamics. It is quite stable since the velocity is implicitly given and consequently it is harder for velocity and position to come out of sync. (As a side note, the well-known demo effect for creating ripples in water uses a similar approach.) It works due to the fact that 2x-x*=x+(x-x*) and x-x* is an approximation of the current velocity (actually, it’s the distance traveled last time step). It is not always very accurate (energy might leave the system, i.e., dissipate) but it’s fast and stable. By lowering the value 2 to something like 1.99 a small amount of drag can also be introduced to the system.

At the end of each step, for each particle the current position x gets stored in the corresponding variable x*. Note that when manipulating many particles, a useful optimization is possible by simply swapping array pointers.

The resulting code would look something like this (the Vector3 class should contain the appropriate member functions and overloaded operators for manipulation of vectors):

// Sample code for physics simulation
class ParticleSystem {
Vector3 m_x[NUM_PARTICLES]; // Current positions
Vector3 m_oldx[NUM_PARTICLES]; // Previous positions
Vector3 m_a[NUM_PARTICLES]; // Force accumulators
Vector3 m_vGravity; // Gravity
float m_fTimeStep;
public:
void TimeStep();
private:
void Verlet();
void SatisfyConstraints();
void AccumulateForces();
// (constructors, initialization etc. omitted)
};
// Verlet integration step
void ParticleSystem::Verlet() {
for(int i=0; i100). As a direct result, the two particles will never come too close to each other. See Figure 8.

Another method for restraining angles is to satisfy a dot product constraint:



Particles can also be restricted to move, for example, in certain planes only. Once again, particles with positions not satisfying the above-mentioned constraints should be moved – deciding exactly how is slightly more complicated that with the stick constraints.

Actually, in Hitman corpses aren’t composed of rigid bodies modeled by tetrahedrons. They are simpler yet, as they consist of particles connected by stick constraints in effect forming stick figures. See Figure 9. The position and orientation for each limb (a vector and a matrix) are then derived for rendering purposes from the particle positions using various cross products and vector normalizations (making certain that knees and elbows bend naturally).

In other words, seen isolated each limb is not a rigid body with the usual 6 degrees of freedom. This means that physically the rotation around the length axis of a limb is not simulated. Instead, the skeletal animation system used to setup the polygonal mesh of the character is forced to orientate the leg, for instance, such that the knee appears to bend naturally. Since rotation of legs and arms around the length axis does not comprise the essential motion of a falling human body, this works out okay and actually optimizes speed by a great deal.

Angular constraints are implemented to enforce limitations of the human anatomy. Simple self collision is taken care of by strategically introducing inequality distance constraints as discussed above, for example between the two knees – making sure that the legs never cross.

For collision with the environment, which consists of triangles, each stick is modeled as a capped cylinder. Somewhere in the collision system, a subroutine handles collisions between capped cylinders and triangles. When a collision is found, the penetration depth and points are extracted, and the collision is then handled for the offending stick in question exactly as described in the beginning of Section 5.

Naturally, a lot of additional tweaking was necessary to get the result just right.

7 Comments
This section contains various remarks that didn’t fit anywhere else.

Motion control
To influence the motion of a simulated object, one simply moves the particles correspondingly. If a person is hit at the shoulder, move the shoulder particle backwards over a distance proportional to the strength of the blow. The Verlet integrator will then automatically set the shoulder in motion.

This also makes it easy for the simulation to ‘inherit’ velocities from an underlying traditional animation system. Simply record the positions of the particles for two frames and then give them to the Verlet integrator, which then automatically continues the motion. Bombs can be implemented by pushing each particle in the system away from the explosion over a distance inversely proportional to the square distance between the particle and the bomb center.

It is possible to constrain a specific limb, say the hand, to a fixed position in space. In this way, one can implement inverse kinematics (IK): Inside the relaxation loop, keep setting the position of a specific particle (or several particles) to the position(s) wanted. Giving the particle infinite mass (invmass=0) helps making it immovable to the physics system. In Hitman, this strategy is used when dragging corpses; the hand (or neck or foot) of the corpse is constrained to follow the hand of the player.

Handling friction
Friction has not been taken care of yet. This means that unless we do something more, particles will slide along the floor as if it were made of ice. According to the Coulomb friction model, friction force depends on the size of the normal force between the objects in contact. To implement this, we measure the penetration depth dp when a penetration has occurred (before projecting the penetration point out of the obstacle). After projecting the particle onto the surface, the tangential velocity vt is then reduced by an amount proportional to dp (the proportion factor being the friction constant). This is done by appropriately modifying x*. See the Figure 10. Care should be taken that the tangential velocity does not reverse its direction – in this case one should simply be set it to zero since this indicates that the penetration point has seized to move tangentially. Other and better friction models than this could and should be implemented.

Collision detection
One of the bottlenecks in physics simulation as presented here lies in the collision detection, which is potentially performed several times inside the relaxation loop. It is possible, however, to iterate a different number of times over the various constraints and still obtain good results.

In Hitman, the collision system works by culling all triangles inside the bounding box of the object simulated (this is done using a octtree approach). For each (static, background) triangle, a structure for fast collision queries against capped cylinders is then constructed and cached. This strategy gave quite a speed boost.

To prevent objects that are moving really fast from passing through other obstacles (because of too large time steps), a simple test if performed. Imagine the line (or a capped cylinder of proper radius) beginning at the position of the object’s midpoint last frame and ending at the position of the object’s midpoint at the current frame. If this line hits anything, then the object position is set to the point of collision. Though this can theoretically give problems, in practice it works fine.

Another collision ‘cheat’ is used for dead bodies. If the unusual thing happens that a fast moving limb ends up being placed with the ends of the capped cylinder on each side of a wall, the cylinder is projected to the side of the wall where the cylinder is connected to the torso.

Miscellaneous
The number of relaxation iterations used in Hitman vary between 1 and 10 with the kind of object simulated. Although this is not enough to accurately solve the global system of constraints, it is sufficient to make motion seem natural. The nice thing about this scheme is that inaccuracies do not accumulate or persist visually in the system causing object drift or the like – in some sense the combination of projection and the Verlet scheme manages to distribute complex calculations over several frames (other schemes have to use further stabilization techniques, like Baumgarte stabilization). Fortunately, the inaccuracies are smallest or even nonexistent when there is little motion and greatest when there is heavy motion – this is nice since fast or complex motion somewhat masks small inaccuracies for the human eye.

A kind of soft bodies can also be implemented by using ‘soft’ constraints, i.e., constraints that are allowed to have only a certain percentage of the deviation ‘repaired’ each frame (i.e., if the rest length of a stick between two particles is 100 but the actual distance is 60, the relaxation code could first set the distance to 80 instead of 100, next frame 90, 95, 97.5 etc.).

As mentioned, we have purposefully refrained from using heavy mathematical notation in order to reach an audience with a broader background. This means that even though the methods presented are firmly based mathematically, their origins may appear somewhat vague or even magical.

For the mathematically inclined, however, what we are doing is actually a sort of time-stepping approach to solving differential inclusions (a variant of differential equations) using a simple sort of interior-point algorithm (see [Stewart] where a similar approach is discussed). When trying to satisfy the constraints, we are actually projecting the system state onto the manifold described by the constraints. This, in turn, is done by solving a system of linear equations. The linear equations or code to solve the constraints can be obtained by deriving the Jacobian of the constraint functions. In this article, relaxation has been discussed as an implicit way of solving the system. Although we haven’t touched the subject here, it is sometimes useful to change the relaxation coefficient or even to use over-relaxation (see [Press] for an explanation). Since relaxation solvers sometimes converge slowly, one might also choose to explicitly construct the equation system and use other methods to solve it (for example a sparse matrix conjugate gradient descent solver with preconditioning using the results from the previous frame (thereby utilizing coherence)).

Note that the Verlet integrator scheme exists in a number of variants, e.g., the Leapfrog integrator and the velocity Verlet integrator. Accuracy might be improved by using these.

Singularities (divisions by zero usually brought about by coinciding particles) can be handled by slightly dislocating particles at random.

As an optimization, bodies should time out when they have fallen to rest.

To toy with the animation system for dead characters in Hitman: Codename 47, open the Hitman.ini file and add the two lines “enableconsole 1” and “consolecmd ip_debug 1” at the bottom. Pointing the cursor at an enemy and pressing shift+F9 will cause a small bomb to explode in his vicinity sending him flying. Press K to toggle free-cam mode (camera is controlled by cursor keys, shift, and ctrl).

Note that since all operations basically take place on the particle level, the algorithms should be very suitable for vector processing (Playstation 2 for example).

8 Conclusion
This paper has described how a physics system was implemented in Hitman. The underlying philosophy of combining iterative methods with a stable integrator has proven to be successful and useful for implementation in computer games. Most notably, the unified particle-based framework, which handles both collisions and contact, and the ability to trade off speed vs. accuracy without accumulating visually obvious errors are powerful features. Naturally, there are still many specifics that can be improved upon. In particular, the tetrahedron model for rigid bodies needs some work. This is in the works.

At IO Interactive, we have recently done some experiments with interactive water and gas simulation using the full Navier-Stokes equations. We are currently looking into applying techniques similar to the ones demonstrated in this paper in the hope to produce faster and more stable water simulation.

9 Acknowledgements
The author wishes to thank Jeroen Wagenaar for fruitful discussions and the entire crew at IO Interactive for cooperation and for producing such a great working environment.

Feedback and comments are very welcome at tj@ioi.dk.

References
[Baraff] Baraff, David, Dynamic Simulation of Non-Penetrating Rigid Bodies, Ph.D. thesis, Dept. of Computer Science, Cornell University, 1992. http://www.cs.cmu.edu/~baraff/papers/index.html
[Mirtich] Mirtich, Brian V., Impulse-base Dynamic Simulation of Rigid Body Systems, Ph.D. thesis, University of California at Berkeley, 1996. http://www.merl.com/people/mirtich/papers/thesis/thesis.html
[Press] Press, William H. et al, Numerical Recipes, Cambridge University Press, 1993. http://www.nr.com/nronline_switcher.html
[Stewart] Stewart, D. E., and J. C. Trinkle, “An Implicit Time-Stepping Scheme for Rigid Body Dynamics with Inelastic Collisions and Coulomb Friction”, International Journal of Numerical Methods in Engineering, to appear. http://www.cs.tamu.edu/faculty/trink/Papers/ijnmeStewTrink.ps
[Verlet] Verlet, L. "Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules", Phys. Rev., 159, 98-103 (1967).
[Witkin] Witkin, Andrew and David Baraff, Physically Based Modeling: Principles and Practice, Siggraph ’97 course notes, 1997. http://www.cs.cmu.edu/~baraff/sigcourse/index.html

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