  We study the filtering problem over a Lie group that plays an important role
in robotics and aerospace applications. We present a new particle filtering
algorithm based on stochastic control. In particular, our algorithm is based on
a duality between smoothing and optimal control. Leveraging this duality, we
reformulate the smoothing problem into an optimal control problem, and by
approximately solving it (using, e.g., iLQR) we establish a superior proposal
for particle smoothing. Combining it with a suitably designed sliding window
mechanism, we obtain a particle filtering algorithm that suffers less from
sample degeneracy compared with existing methods. The efficacy of our algorithm
is illustrated by a filtering problem over SO(3) for satellite attitude
estimation.
