Mathematics, Systems and Robotics Seminar 

23/11/2007, 15:00 — 16:00 — Room P10, Mathematics Building
Rita Cunha and Carlos Silvestre, ISR

Output feedback control for stabilization on SE(3)

We address the problem of stabilizing systems that evolve on the Special Euclidean Group $\mathrm{SE}(3)$. The proposed solution consists of an output-feedback controller that guarantees almost global asymptotic stability (GAS) of the desired equilibrium point, in the sense that the equilibrium point is stable and we have convergence for all initial conditions except for those in a set of measure zero. The output vector is formed by the position coordinates, expressed in the body frame, of a collection of landmarks fixed in the environment. As such, the method can be used to perform a variety of robotic tasks such as positioning a manipulator’s end-effector with respect to an object to be grasped or landing an UAV over a predefined target using onboard sensor measurements. The resulting closed-loop system exhibits the following properties:

1. the position and orientation subsystems are decoupled,
2. the position error is globally exponentially stable, and
3. the orientation error is almost globally exponentially stable.

Results are also provided that allow one to select landmark configurations so as to control how the position and orientation of the rigid body converge to their final equilibrium values. Building on these results, a vision-based controller is proposed that not only guarantees almost GAS of the target configuration, but also ensures the positive invariance of a subset of $\mathrm{SE}(3)$, specifically defined to enforce feature visibility throughout the closed-loop trajectories of the camera.