Soft Constraints

We must also integrate image observation data into the time evolution of the dynamic model. To accomplish this we use the observation model developed in our earlier stereo blob-tracking work (briefly described below), which relates the 2-D distribution of pixel values to a tracked object's 3-D position and orientation, thus providing a bridge to the physical modeling layer. Because these observation data are noisy, they influence the dynamic model but do not impose hard constraints on its behavior. Consequently, observation data provide a sort of soft constraint on the model, and may be thought of as forces acting on the model's dynamics. For a large class of Kalman filters this analogy is exact, that is, the mathematics of integrating forces acting on a dynamic body is identical to the mathematics of observation integration within a Kalman filter [16,13,4]. Soft constraints have the additional advantage that they can also be used to model external influences on the dynamic system, such as gravity.

Soft constraints such as these can be expressed as a potential field acting on the dynamic system. A potential field is function over space that, when evaluated at a given position, applies a force to the model:

$${\bf Q}_f = f({\bf p})$$

Where ${\bf Q}_f$ is the component of ${\bf Q}$ contributed by the potential field when measured at point ${\bf p}$. The potential field is a good abstraction for modeling data since typical sensor noise distributions can be easily modeled as deformations of the field, for instance, measurement uncertainty in one direction can be modeled by making the potential field proportionally broader or narrower. The incorporation of a potential field function that models a probability density pushes the dynamic evolution of the model toward the most likely value, starting from the current model state.

Note that functions that take the model state as input, such as a control law, can also be represented as a time-varying potential field. One relevant example is incorporation of a probability distribution over link position and velocity:

$${\bf Q}_f = f({\bf p}, {\bf q} , \dot{\bf q})$$