  Determining the reachable set for a given nonlinear system is critically
important for autonomous trajectory planning for reach-avoid applications and
safety critical scenarios. Providing the reachable set is generally impossible
when the dynamics are unknown, so we calculate underapproximations of such sets
using local dynamics at a single point and bounds on the rate of change of the
dynamics determined from known physical laws. Motivated by scenarios where an
adverse event causes an abrupt change in the dynamics, we attempt to determine
a provably reachable set of states without knowledge of the dynamics. This
paper considers systems which are known to operate on a manifold.
Underapproximations are calculated by utilizing the aforementioned knowledge to
derive a guaranteed set of velocities on the tangent bundle of a complete
Riemannian manifold that can be reached within a finite time horizon. We then
interpret said set as a control system; the trajectories of this control system
provide us with a guaranteed set of reachable states the unknown system can
reach within a given time. The results are general enough to apply on systems
that operate on any complete Riemannian manifold. To illustrate the practical
implementation of our results, we apply our algorithm to a model of a pendulum
operating on a sphere and a three-dimensional rotational system which lives on
the abstract set of special orthogonal matrices.
