Liu Huan, Ge Qiaode Jeffrey, Langer Mark P
Computational Design Kinematics Lab, Stony Brook University, SUNY, Stony Brook, New York, 11794-2300.
Radiation Oncology Department, Indiana University, Indianapolis, Indiana, 46202.
Proc ASME Des Eng Tech Conf. 2023 Aug;8(47). doi: 10.1115/detc2023-116793. Epub 2023 Nov 21.
Kinematics is most commonly about the motion of unbounded spaces. This paper deals with the kinematics of bounded shapes in a plane. This paper studies the problem of motion interpolation of a planar object with its shape taken into consideration. It applies and extends a shape dependent distance measure between two positions in the context of motion interpolation. Instead of using a fixed reference frame, a shape-dependent inertia frame of reference is used for formulating the distance between positions of a rigid object in a plane. The resulting distance function is then decomposed in two orthogonal directions and is used to formulate an interpolating function for the distance functions in these two directions. This leads to a shape dependent interpolation of translational components of a planar motion. In difference to the original concept of Kazerounian and Rastegar that comes with a shape dependent measure of the angular motion, it is assumed in this paper that the angular motion is shape independent as the angular metric is dimensionless. The resulting distance measure is not only a combination of translation and rotation parameters but also depends on the area moments of inertia of the object. It derives the explicit expressions for decomposing the shape dependent distance in two orthogonal directions, which is then used to obtain shape dependent motion interpolants in these directions. The resulting interpolants have similarities to the well-known spherical linear interpolants widely used in computer graphics in that they are defined using sinusoidal functions instead of linear interpolation in Euclidean space. The path of the interpolating motion can be adjusted by different choice of shape parameters. Examples are provided to illustrate the effect of object shapes on the resulting interpolating motions.
运动学通常研究的是无界空间的运动。本文探讨的是平面中有界形状的运动学问题。本文研究了在考虑平面物体形状的情况下进行运动插值的问题。在运动插值的背景下，本文应用并扩展了一种依赖于形状的两个位置之间的距离度量。不再使用固定参考系，而是使用依赖于形状的惯性参考系来确定平面中刚体两个位置之间的距离。然后将得到的距离函数在两个正交方向上分解，并用于构建这两个方向上距离函数的插值函数。这就得到了平面运动平移分量的依赖于形状的插值。与卡泽鲁尼安和拉斯特加最初提出的带有依赖于形状的角运动度量的概念不同，本文假设角运动与形状无关，因为角度量是无量纲的。得到的距离度量不仅是平移和旋转参数的组合，还取决于物体的面积惯性矩。本文推导了在两个正交方向上分解依赖于形状的距离的显式表达式，然后用于获得这些方向上依赖于形状的运动插值。得到的插值与计算机图形学中广泛使用的著名球面线性插值有相似之处，即它们是用正弦函数定义的，而不是欧几里得空间中的线性插值。通过不同的形状参数选择可以调整插值运动的路径。文中给出了示例来说明物体形状对所得插值运动的影响。
