  In this paper, we formulate a novel trajectory optimization scheme that takes
into consideration the state uncertainty of the robot and obstacle into its
collision avoidance routine. The collision avoidance under uncertainty is
modeled here as an overlap between two distributions that represent the state
of the robot and obstacle respectively. We adopt the minmax procedure to
characterize the area of overlap between two Gaussian distributions, and
compare it with the method of Bhattacharyya distance. We provide closed form
expressions that can characterize the overlap as a function of control. Our
proposed algorithm can avoid overlapping uncertainty distributions in two
possible ways. Firstly when a prescribed overlapping area that needs to be
avoided is posed as a confidence contour lower bound, control commands are
accordingly realized through a MPC framework such that these bounds are
respected. Secondly in tight spaces control commands are computed such that the
overlapping distribution respects a prescribed range of overlap characterized
by lower and upper bounds of the confidence contours. We test our proposal with
extensive set of simulations carried out under various constrained
environmental configurations. We show usefulness of proposal under tight spaces
where finding control maneuvers with minimal risk behavior becomes an
inevitable task.
