Implementing virtual fixtures in guiding tasks constrains the movement of the robot's end effector to specific curves within its workspace. However, incorporating guiding frameworks may encounter discontinuities when optimizing the reference target position to the nearest point relative to the current robot position. This article aims to give a geometric interpretation of such discontinuities, with specific reference to the commonly adopted Gauss-Newton algorithm. The effect of such discontinuities, defined as Euclidean Distance Singularities, is experimentally proved. We then propose a solution that is based on a Linear Quadratic Tracking problem with minimum jerk command, then compare and validate the performances of the proposed framework in two different human-robot interaction scenarios.

翻译：在引导任务中实现虚拟夹具会限制机器人末端执行器在工作空间内沿特定曲线运动。然而，当将参考目标位置优化为相对于当前机器人位置最近点时，引入引导框架可能遭遇不连续性。本文旨在对此类不连续性给出几何解释，特别针对广泛采用的高斯-牛顿算法。我们通过实验证明了此类不连续性（定义为欧几里得距离奇异性）的影响。随后提出一种基于最小加加速度指令的线性二次跟踪问题解决方案，并在两种不同人机交互场景中比较和验证了所提框架的性能。