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The purpose of this paper is to preset a spherical element subdivision method for the numerical evaluation of nearly singular integrals in three-dimensional (3D) boundary element method (BEM).

In this method, the source point is first projected to the tangent plane of the element. Then two cases are considered: the projection point is either inside or outside the element. In both cases, the element is subdivided into a number of patches using a sequence of spheres with decreasing radius.

With the proposed method, the patches obtained are automatically refined as they approach the projection point and each patch of the integration element is “good” in shape and size for standard Gaussian quadrature. Therefore, all kinds of nearly singular boundary integrals on elements of any shape and size with arbitrary source point location related to the element can be evaluated accurately and efficiently.

Numerical examples for planar and slender elements with various relative location of the source point are presented. The results demonstrate that our method has much better accuracy, efficiency and stability than conventional methods.

This study aims to propose a virtual fixture (VF) – assisted robot teleoperation framework that modulates the generated trajectory from demonstrations to response varying obstacles in complex environment.

First, a single trajectory is learned from demonstration using dynamic movement primitives (DMP). Then, the classic DMP is improved by integrating adaptive terms and updating the radial basis function kernel weights, allowing the single-skill trajectory to alter to respond to obstacles dynamically. Finally, composite virtual forces are generated on the haptic device to enhance operational stability and prevents unintentional operations that could bring the robot into close proximity with obstacles.

The VF method can accomplish online obstacle avoidance and operation guidance. The experimental results show that the operation trajectory based on VFs is smoother compared to the operation without assistance. Moreover, the operational speed has increased by up to 44.37% compared to the demonstrations.

The proposed composite VF-based protection framework solves the problem that classical DMP method cannot dynamically avoid obstacles, and effectively improves the operational safety and efficiency.