Koopman Operator Based Modeling and Control of Rigid Body Motion Represented by Dual Quaternions
Koopman Operator Based Modeling and Control of Rigid Body Motion Represented by Dual Quaternions
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DOI：
10.23919/acc53348.2022.9867584
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发表时间：
2021-10
期刊：
影响因子：
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通讯作者：
Vrushabh Zinage;E. Bakolas
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文献类型：
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作者：
Vrushabh Zinage;E. Bakolas
在本文中，我们系统地推导出一个有限的Koopman基于观测量构建一个提升的线性状态空间模型，描述了基于对偶四元数表示的刚体动力学。扩展动态模式分解（EDMD）等方法可以计算不同类型问题的Koopman算子的有限近似，但通常情况下，它们不能保证非线性动力学的计算近似足够准确，除非有一组适当的可观测量。该领域最先进的方法通过使用神经网络、标准径向基函数（RBF）、这些函数的多项式或启发式近似来计算可观察量的近似。然而，这些观测值可能不会产生足够精确的动力学近似。与此相反，我们首先证明了导出的可观测函数的逐点收敛到零。接下来，我们使用EDMD中导出的观测值来计算刚体动力学的提升线性状态和输入矩阵。最后，我们表明，LQR型（线性）控制器，这是截断的线性状态空间模型的基础上设计的，可以引导刚体到所需的状态，而其性能是相称的非线性控制器。我们的方法的有效性通过数值模拟证明。
In this paper, we systematically derive a finite set of Koopman based observables to construct a lifted linear state space model that describes the rigid body dynamics based on the dual quaternion representation. Methods such as the Extended Dynamic Mode Decomposition (EDMD) can compute finite approximations of the Koopman operator for different classes of problems but in general, they cannot offer guarantees that the computed approximation of the nonlinear dynamics is sufficiently accurate unless an appropriate set of observables is available. State-of-the-art methods in the field compute approximations of the observables by using neural networks, standard radial basis functions (RBFs), polynomials or heuristic approximations of these functions. However, these observables might not yield a sufficiently accurate approximation of the dynamics. In contrast, we first show the pointwise convergence of the derived observable functions to zero. Next, we use the derived observables in EDMD to compute the lifted linear state and input matrices for the rigid body dynamics. Finally, we show that an LQR type (linear) controller, which is designed based on the truncated linear state space model, can steer the rigid body to a desired state while its performance is commensurate with that of a nonlinear controller. The efficacy of our approach is demonstrated through numerical simulations.