Almost-global tracking of simple mechanical systems on a general class of Lie Groups
Almost-global tracking of simple mechanical systems on a general class of Lie Groups
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DOI：
10.1109/tac.2005.862219
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发表时间：
2006-02
影响因子：
6.8
通讯作者：
D. H. S. Maithripala;J. Berg;W. Dayawansa
中科院分区：
文献类型：
--
作者：
D. H. S. Maithripala;J. Berg;W. Dayawansa
当位形空间是一类一般李群时，我们给出了一种完全驱动简单机械系统的一般本征跟踪控制器设计。我们首先将状态反馈控制器表示为满足某些正则性条件的函数--“误差函数”。如果可以找到误差函数，则可以从几乎每个初始条件渐近跟踪一般光滑且有界的参考轨迹，并且具有局部指数收敛。几乎所有初始条件的渐近收敛称为“几乎全局”渐近稳定性。可以证明误差函数存在于任何紧李群上，或者存在于紧李群和R/sup n/的乘积上的任何李群上。这涵盖了许多实际感兴趣的情况，例如SO(N)、SE(N)、它们的子群和直积。我们证明了对于紧李群，由全状态反馈律和指数收敛的速度观测器组成的动态构形反馈控制器对于跟踪误差也是几乎全局渐近稳定的。我们强调，这些结果不需要不变性。然而，对于动能是左不变的特殊情况，我们证明了这些控制器的显式表示不需要李群上的坐标。在SO(3)上演示了控制器的结构，并对轴对称陀螺进行了仿真。结果表明，该算法具有较好的性能。
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.