Collision intensive and multi-body simulations are difficult to constrain because they exhibit extreme sensitivity to initial conditions or other simulation parameters, which has restricted their use in applications. For instance, a feature film might require a scene in which dice are shown on a table, only to be knocked onto the floor to come up showing snake eyes in a specific location (an example due to Barzel, Hughes and Wood).

We have developed a methodology for solving such problems that begins by adding plausible sources of uncertainty to the model to be constrained. A probability distribution is then defined for the model which assigns higher probability to desirable animations that are physically plausible, and low probability to unsatisfactory animations or those that are unlikely in the world. A Markov chain Monte Carlo (MCMC) algorithm may then be used to sample animations according to the distribution. Other benefits come from this approach:

- The real world contains fine scale variation that traditional simulation models generally ignore. Our technique can capture this variation. In training environments, this results in the subject developing skills more compatible with the real world: a driver trained on simulations of bumpy roads will be better prepared for real world road surfaces.
- Visually, procedural animations can be more believable when uncertainty is added.
- Sampling with an MCMC algorithm ensures that the animations we generate are properly distributed according to the model, so we know that the solutions generated are plausible.
- In a world with uncertainty, we generally expect a constrained problem to have multiple solutions. It is difficult to know beforehand what solutions are available, which compounds any difficulties a user may have in codifying their preferences. With a sampling approach, it is possible to explore the range of solutions available without a user explicitly stating what they desire.
- Applications like training simulations and computer games benefit directly from the generation of multiple solutions through sampling.

We have demonstrated this technique with many examples of collision intensive multi-body simulations that may be constrained to give a specific outcome. No other technique has been described that can handle such difficult systems and hard to satisfy constraints.

Stephen Chenney and D.A.Forsyth, "Sampling Plausible Solutions to Multi-Body Constraint Problems". SIGGRAPH 2000 Conference Proceedings, pages 219-228, July 2000.