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On the global interpolation of motion

机译：关于运动的全局插值

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Interpolation of motion is required in various fields of engineering such as computer animation and vision, trajectory planning for robotics, optimal control of dynamical systems, or finite element analysis. While interpolation techniques in the Euclidean space are well established, general approaches to interpolation on manifolds remain elusive. Interpolation schemes in the Euclidean space can be recast as minimization problems for weighted distance metrics. This observation allows the straightforward generalization of interpolation in the Euclidean space to interpolation on manifolds, provided that a metric of the manifold is defined. This paper proposes four metrics of the motion manifold: the matrix, quaternion, vector, and geodesic metrics. For each of these metrics, the corresponding interpolation schemes are derived and their advantages and drawbacks are discussed. It is shown that many existing interpolation schemes for rotation and motion can be derived from the minimization framework proposed here. The problems of averaging of rotation and motion can be treated easily within the same framework. Both local and global interpolation problems are addressed. The proposed interpolation framework can be used with any suitable set of basis functions. Examples are presented with Chebyshev spectral, Fourier spectral, and B-spline basis functions. This paper also introduces one additional approach to the interpolation of motion based on the interpolation of its derivatives. While this approach provides high accuracy, the associated computational cost is high and the approach cannot be used in multi-variable interpolation easily. (C) 2018 Elsevier B.V. All rights reserved.
机译：在各种工程领域中都需要进行运动插补，例如计算机动画和视觉，机器人技术的轨迹规划，动态系统的最佳控制或有限元分析。尽管在欧几里得空间中的插值技术已经很好地建立了，但是在流形上进行插值的一般方法仍然难以捉摸。欧几里得空间中的插值方案可以重铸为加权距离度量的最小化问题。如果定义了流形的度量，则此观察结果可以将欧氏空间中的插值直接推广到流形上的插值。本文提出了运动流形的四个度量：矩阵，四元数，向量和测地度量。对于这些指标中的每一个，都推导了相应的插值方案，并讨论了它们的优缺点。结果表明，许多现有的旋转和运动插值方案都可以从此处提出的最小化框架中得出。旋转和运动平均的问题可以在同一框架内轻松解决。局部和全局插值问题均得到解决。所提出的插值框架可以与任何适当的基础函数集一起使用。用切比雪夫谱，傅里叶谱和B样条基函数给出了示例。本文还介绍了一种基于运动导数插值的运动插值方法。尽管该方法提供了高精度，但是相关的计算成本很高，并且该方法不能容易地用于多变量插值中。 （C）2018 Elsevier B.V.保留所有权利。

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