Qiaode Jeffrey Ge, Zihan Yu, Mona Arbab, Mark P Langer
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引用次数: 0
Abstract
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.
期刊介绍:
Fundamental theory, algorithms, design, manufacture, and experimental validation for mechanisms and robots; Theoretical and applied kinematics; Mechanism synthesis and design; Analysis and design of robot manipulators, hands and legs, soft robotics, compliant mechanisms, origami and folded robots, printed robots, and haptic devices; Novel fabrication; Actuation and control techniques for mechanisms and robotics; Bio-inspired approaches to mechanism and robot design; Mechanics and design of micro- and nano-scale devices.