  This paper introduces an extension of the LQR-tree algorithm, which is a
feedback-motion-planning algorithm for stabilizing a system of ordinary
differential equations from a bounded set of initial conditions to a goal. The
constructed policies are represented by a tree of exemplary system
trajectories, so called demonstrations, and linear-quadratic regulator (LQR)
feedback controllers. Consequently, the crucial component of any LQR-tree
algorithm is a demonstrator that provides suitable demonstrations. In previous
work, such a demonstrator was given by a local trajectory optimizer. However,
these require appropriate initial guesses of solutions to provide valid
results, which was pointed out, but largely unresolved in previous
implementations. In this paper, we augment the LQR-tree algorithm with a
randomized motion-planning procedure to discover new valid demonstration
candidates to initialize the demonstrator in parts of state space not yet
covered by the LQR-tree. In comparison to the previous versions of the LQR-tree
algorithm, the resulting exploring LQR-tree algorithm reliably synthesizes
feedback control laws for a far more general set of problems.
