  In this paper, we propose an approach to designing online feedback
controllers for input-saturated robotic systems evolving on Lie groups by
extending the recently developed Sequential Action Control (SAC). In contrast
to existing feedback controllers, our approach poses the nonconvex constrained
nonlinear optimization problem as the tracking of a desired negative mode
insertion gradient on the configuration space of a Lie group. This results in a
closed-form feedback control law even with input saturation and thus is well
suited for online application. In extending SAC to Lie groups, the associated
mode insertion gradient is derived and the switching time optimization on Lie
groups is studied. We demonstrate the efficacy and scalability of our approach
in the 2D kinematic car on SE(2) and the 3D quadrotor on SE(3). We also
implement iLQG on a quadrator model and compare to SAC, demonstrating that SAC
is both faster to compute and has a larger basin of attraction.
