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首页> 外文期刊>IEEE Transactions on Automatic Control >Almost-Global Tracking of Simple Mechanical Systems on a General Class of Lie Groups
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Almost-Global Tracking of Simple Mechanical Systems on a General Class of Lie Groups

机译：一般李群上的简单机械系统的几乎全局跟踪

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摘要

We present a general intrinsic tracking controller design for fully-actuated simple mechanical systems, when the configuration space is one of a general class of Lie groups. We first express a state-feedback controller in terms of a function - the "error function" - satisfying certain regularity conditions. If an error function can be found, then a general smooth and bounded reference trajectory may be tracked asymptotically from almost every initial condition, with locally exponential convergence. Asymptotic convergence from almost every initial condition is referred to as "almost-global" asymptotic stability. Error functions may be shown to exist on any compact Lie group, or any Lie group diffeomorphic to the product of a compact Lie group and R{sup}n. This covers many cases of practical interest, such as SO(n), SE(n), their subgroups, and direct products. We show here that for compact Lie groups the dynamic configuration-feedback controller obtained by composing the full state-feedback law with an exponentially convergent velocity observer is also almost-globally asymptotically stable with respect to the tracking error. We emphasize that no invariance is needed for these results. However, for the special case where the kinetic energy is left-invariant, we show that the explicit expression of these controllers does not require coordinates on the Lie group. The controller constructions are demonstrated on SO(3), and simulated for the axi-symmetric top. Results show excellent performance.
机译：当配置空间是李群的一般类别之一时，我们提出了一种用于全驱动简单机械系统的一般内在跟踪控制器设计。我们首先用满足某些规则性条件的函数-“错误函数”来表示状态反馈控制器。如果可以找到误差函数，则可以从几乎每个初始条件渐近地跟踪一般平滑且有界的参考轨迹，并且具有局部指数收敛性。来自几乎每个初始条件的渐近收敛被称为“几乎全局”渐近稳定性。可能显示出误差函数存在于任何紧凑的Lie基团上，或对紧凑的Lie基团与R {sup} n乘积有差别的任何Lie基团。这涵盖了许多具有实际意义的案例，例如SO（n），SE（n），它们的子组和直接乘积。我们在这里表明，对于紧凑的李群，通过将全状态反馈定律与指数收敛速度观测器组成而获得的动态配置反馈控制器在跟踪误差方面也几乎是全局渐近稳定的。我们强调，这些结果不需要不变性。但是，对于动能为左不变的特殊情况，我们证明了这些控制器的显式表达式不需要Lie组上的坐标。控制器构造在SO（3）上进行了演示，并针对轴对称顶部进行了仿真。结果显示出优异的性能。

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