  Robotic systems often need to consider multiple tasks concurrently. This
challenge calls for controller synthesis algorithms that fulfill multiple
control specifications while maintaining the stability of the overall system.
In this paper, we decompose multi-objective tasks into subtasks, where
individual subtask controllers are designed independently and then combined to
generate the overall control policy. In particular, we adopt Riemannian Motion
Policies (RMPs), a recently proposed controller structure in robotics, and,
RMPflow, its associated computational framework for combining RMP controllers.
We re-establish and extend the stability results of RMPflow through a rigorous
Control Lyapunov Function (CLF) treatment. We then show that RMPflow can stably
combine individually designed subtask controllers that satisfy certain CLF
constraints. This new insight leads to an efficient CLF-based computational
framework to generate stable controllers that consider all the subtasks
simultaneously. Compared with the original usage of RMPflow, our framework
provides users the flexibility to incorporate design heuristics through nominal
controllers for the subtasks. We validate the proposed computational framework
through numerical simulation and robotic implementation.
