Structured linearization of discrete mechanical systems on Lie groups: A synthesis of analysis and control
Structured linearization of discrete mechanical systems on Lie groups: A synthesis of analysis and control
复制标题
李群上离散机械系统的结构化线性化：分析与控制的综合
DOI：
10.1109/cdc.2015.7402357
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
T. Murphey
中科院分区：
文献类型：
--
作者：
Taosha Fan;T. Murphey
李群变分积分器具有变分积分器和李群积分器的优点，它保持了动量，辛形式，完整约束和李群结构。此外，它们的长时间能量稳定行为和坐标独立性使其非常适合模拟各种机械系统。李群变分积分的保结构性意味着它的线性化也是保结构的，因此我们称这种线性化为“结构线性化”。然而，由于变分积分的隐式性质和李群的非平凡微分结构，使得李群变分积分的结构线性化比广义坐标下的线性化复杂得多。在本文中，我们制定了李群变分积分的结构线性化综合现有的分析和控制工具。为了说明本文的实用性，直接在约束李群上构造LQR控制器的非对称三维单摆和四旋翼悬挂负载，仿真结果表明，这两种控制器具有较大的吸引域。
Lie group variational integrators have the advantages of both variational and Lie group integrators, which preserve the momentum, symplectic form, holonomic constraints and the Lie group structure. In addition, their long-time energy stable behaviour and coordinate-independent nature make it quite suitable to simulate a variety of mechanical systems. The structure-preservation of a Lie group variational integrator implies its linearization is structure-preserving as well, thus we call such a linearization “structured linearization”. However, due to the implicit nature of variational integrators and the non-trivial differential structure of Lie groups, the structured linearization of Lie group variational integrators is much more complicated than that in generalized coordinates. In this paper, we formulate the structured linearization of Lie group variational integrators to synthesize existing analysis and control tools. To illustrate the utility of the paper, LQR controllers are constructed directly on constrained Lie groups for the asymmetric 3D pendulum and quadrotor with a suspended load, simulation results show that both controllers have a large basin of attraction.