Mechanical systems naturally evolve on principal bundles describing their inherent symmetries. The ensuing factorization of the configuration manifold into a symmetry group and an internal shape space has provided deep insights into the locomotion of many robotic and biological systems. On the other hand, the property of differential flatness has enabled efficient, effective planning and control algorithms for various robotic systems. Yet, a practical means of finding a flat output for an arbitrary robotic system remains an open question. In this work, we demonstrate surprising new connections between these two domains, for the first time employing symmetry directly to construct a flat output. We provide sufficient conditions for the existence of a trivialization of the bundle in which the group variables themselves are a flat output. We call this a geometric flat output, since it is equivariant (i.e. maintains the symmetry) and is often global or almost-global, properties not typically enjoyed by other flat outputs. In such a trivialization, the motion planning problem is easily solved, since a given trajectory for the group variables will fully determine the trajectory for the shape variables that exactly achieves this motion. We provide a partial catalog of robotic systems with geometric flat outputs and worked examples for the planar rocket, planar aerial manipulator, and quadrotor.

翻译：机械系统自然地在描述其固有对称性的主丛上演化。构型流形由此分解为对称群与内部形状空间，为理解许多机器人与生物系统的运动提供了深刻见解。另一方面，微分平坦性为各类机器人系统的高效规划与控制算法提供了支持。然而，为任意机器人系统寻找平坦输出的实用方法仍是一个开放问题。本文揭示了这两个领域间令人惊奇的新联系，首次直接利用对称性构建平坦输出。我们给出了丛的平凡化存在的充分条件，使得群变量本身成为平坦输出。我们将其称为几何平坦输出，因为它具有等变性（即保持对称性）且通常是全局或几乎全局的——这些特性是其他平坦输出所不具备的。在此类平凡化下，运动规划问题得以轻松解决：给定群变量的轨迹将完全确定精确实现该运动的形状变量轨迹。我们提供了部分具有几何平坦输出的机器人系统目录，并给出了平面火箭、平面空中机械臂和四旋翼飞行器的计算示例。