  Despite decades of work in fast reactive planning and control, challenges
remain in developing reactive motion policies on non-Euclidean manifolds and
enforcing constraints while avoiding undesirable potential function local
minima. This work presents a principled method for designing and fusing desired
robot task behaviors into a stable robot motion policy, leveraging the
geometric structure of non-Euclidean manifolds, which are prevalent in robot
configuration and task spaces. Our Pullback Bundle Dynamical Systems (PBDS)
framework drives desired task behaviors and prioritizes tasks using separate
position-dependent and position/velocity-dependent Riemannian metrics,
respectively, thus simplifying individual task design and modular composition
of tasks. For enforcing constraints, we provide a class of metric-based tasks,
eliminating local minima by imposing non-conflicting potential functions only
for goal region attraction. We also provide a geometric optimization problem
for combining tasks inspired by Riemannian Motion Policies (RMPs) that reduces
to a simple least-squares problem, and we show that our approach is
geometrically well-defined. We demonstrate the PBDS framework on the sphere
$\mathbb S^2$ and at 300-500 Hz on a manipulator arm, and we provide task
design guidance and an open-source Julia library implementation. Overall, this
work presents a fast, easy-to-use framework for generating motion policies
without unwanted potential function local minima on general manifolds.
