Abstract
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.
Abstract (translated)
本文提出了一种新的向量场策略，用于在连接矩阵李群上的完全驱动系统进行控制，确保能够收敛到并沿着该群上定义的曲线行进。我们的方法扩展了我们先前的工作（Rezende等人，2022年），并在考虑欧几里得空间中的平移李群时退化为前者。由于Rezende等人的证明（2022）依赖于诸如收敛分量和遍历分量之间正交性之类的特性，我们在本文中通过利用李群的属性来扩展这些结果。这些性质还允许控制输入不冗余，即其维度与李群的维度相匹配，而不仅仅是该群嵌入的空间可能更大的维度。在某些情况下，这可以导致更实用的控制输入。我们策略的一个特别值得注意的应用是在SE(3)上的系统控制——在这种情况下，非冗余输入对应于物体的机械扭动——使其非常适合用于可以自由移动和旋转的对象，例如全方位无人机。在这个情况下，我们提供了一种计算向量场的有效算法，并通过使用机器人操作器进行实验验证来展示该方法的效果。
URL
https://arxiv.org/abs/2602.21450
