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In state estimation, we often want the maximum likelihood estimate of the current state. For the commonly used joint multivariate Gaussian distribution over the state space, this can be efficiently found using a Kalman filter. However, in complex environments the state space is often highly constrained. For example, for objects within a refrigerator, they cannot interpenetrate each other or the refrigerator walls. The multivariate Gaussian is unconstrained over the state space and cannot incorporate these constraints. In particular, the state estimate returned by the unconstrained distribution may itself be infeasible. Instead, we solve a related constrained optimization problem to find a good feasible state estimate. We illustrate this for estimating collision-free configurations for objects resting stably on a 2-D surface, and demonstrate its utility in a real robot perception domain.
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