Browsing by Author "Ge, Qiaode Jeffrey"

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    On the Computation of Mean and Variance of Spatial Displacements
    (ASME, 2024) Ge, Qiaode Jeffrey; Yu, Zihan; Arbab, Mona; Langer, Mark P.; Radiation Oncology, School of Medicine
    This paper studies the problem of computing an average (or mean) displacement from a set of given spatial displacements using three types of parametric representations: Euler angles and translation vectors, unit quaternions and translation vectors, and dual quaternions. It is shown that the use of Euclidean norm in the space of unit quaternions reduces the problem to that of computing the mean for each quaternion component separately and independently. While the resulting algorithm is simple, a change in the sign of a unit quaternion could lead to an incorrect result. A novel kinematic measure based on dual quaternions is introduced to capture the separation between two spatial displacements. This kinematic measure is used to define the variance of a set of displacements, which is then used to formulate a constrained least squares minimization problem. It is shown that the problem decomposes into that of finding the optimal translation vector and the optimal unit quaternion. The former is simply the centroid of the set of translation vectors and the latter is obtained as the eigenvector corresponding to the least eigenvalue of a 4 × 4 positive definite symmetric matrix. In addition, it is found that the weight factor used in combining rotations and translations in the formulation does not play a role in the final outcome. Examples are provided to show the comparisons of these methods.
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    On the Construction of Confidence Regions for Uncertain Planar Displacements
    (ASME, 2024) Yu, Zihan; Ge, Qiaode Jeffrey; Langer, Mark P.; Arbab, Mona; Radiation Oncology, School of Medicine
    This paper studies the statistical concept of confidence region for a set of uncertain planar displacements with a certain level of confidence or probabilities. Three different representations of planar displacements are compared in this context and it is shown that the most commonly used representation based on the coordinates of the moving frame is the least effective. The other two methods, namely the exponential coordinates and planar quaternions, are equally effective in capturing the group structure of SE(2). However, the former relies on the exponential map to parameterize an element of SE(2), while the latter uses a quadratic map, which is often more advantageous computationally. This paper focus on the use of planar quaternions to develop a method for computing the confidence region for a given set of uncertain planar displacements. Principal component analysis (PCA) is another tool used in our study to capture the dominant direction of movements. To demonstrate the effectiveness of our approach, we compare it to an existing method called rotational and translational confidence limit (RTCL). Our examples show that the planar quaternion formulation leads to a swept volume that is more compact and more effective than the RTCL method, especially in cases when off-axis rotation is present.
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    On the Construction of Kinematic Confidence Ellipsoids for Uncertain Spatial Displacements
    (Springer, 2023) Yu, Zihan; Ge, Qiaode Jeffrey; Langer, Mark P.; Arbab, Mona; Radiation Oncology, School of Medicine
    This paper deals with the problem of estimating confidence regions of a set of uncertain spatial displacements for a given level of confidence or probabilities. While a direct application of the commonly used statistic methods to the coordinates of the moving frame is straight-forward, it is also the least effective in that it grossly overestimate the confidence region. Based on the dual-quaternion representation, this paper introduces the notion of the kinematic confidence ellipsoids as an alternative to the existing method called rotation and translation confidence limit (RTCL). An example is provided to demonstrate how the kinematic confidence ellipsoids can be computed.
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    Shape Dependent Motion Interpolants for Planar Objects
    (ASME, 2023) Liu, Huan; Ge, Qiaode Jeffrey; Langer, Mark P.; Radiation Oncology, School of Medicine
    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.