Constructive Vector Fields for Path Following in Fully-Actuated Systems on Matrix Lie Groups

This paper presents a novel vector field strategy for controlling fully-actuated systems on connected matrix Lie groups, ensuring convergence to and traversal along a curve defined on the group. Our approach generalizes our previous work (Rezende et al., 2022) and reduces to it when considering the Lie group of translations in Euclidean space. Since the proofs in Rezende et al. (2022) rely on key properties such as the orthogonality between the convergent and traversal components, we extend these results by leveraging Lie group properties. These properties also allow the control input to be non-redundant, meaning it matches the dimension of the Lie group, rather than the potentially larger dimension of the space in which the group is embedded. This can lead to more practical control inputs in certain scenarios. A particularly notable application of our strategy is in controlling systems on SE(3) -- in this case, the non-redundant input corresponds to the object's mechanical twist -- making it well-suited for controlling objects that can move and rotate freely, such as omnidirectional drones. In this case, we provide an efficient algorithm to compute the vector field. We experimentally validate the proposed method using a robotic manipulator to demonstrate its effectiveness.