The control of free-floating robots requires dealing with several challenges. The motion of such robots evolves on a continuous manifold described by the Special Euclidean Group of dimension 3, known as SE(3). Methods from finite horizon Linear Quadratic Regulators (LQR) control have gained recent traction in the robotics community. However, such approaches are inherently solving an unconstrained optimization problem and hence are unable to respect the manifold constraints imposed by the group structure of SE(3). This may lead to small errors, singularity problems and double cover issues depending on the choice of coordinates to model the floating base motion. In this paper, we propose the use of canonical exponential coordinates of SE(3) and the associated Exponential map along with its differentials to embed this structure in the theory of finite horizon LQR controllers.

翻译：自由漂浮机器人的控制需要应对多项挑战。其运动定义在由三维特殊欧氏群（Special Euclidean Group，简称SE(3))描述的连续流形上。有限时域线性二次型调节器（Linear Quadratic Regulators，LQR）控制方法近年来在机器人学界备受关注。然而，此类方法本质上是求解无约束优化问题，因此无法满足SE(3)群结构施加的流形约束。根据漂浮基座运动建模时采用的坐标系选择，这可能导致微小误差、奇异性问题以及双重覆盖难题。本文提出采用SE(3)的典型指数坐标及其伴随的指数映射及其微分，将该结构嵌入有限时域LQR控制器理论中。