A configuration of a mechanical linkage is defined as regular if there exists a subset of actuators with their corresponding Jacobian columns spans the gripper's velocity space. All other configurations are defined in the literature as singular configurations. Consider mechanisms with grippers' velocity space m. We focus our attention on the case where m Jacobian columns of such mechanism span m, while all the rest are linearly dependent. These are obviously an undesirable configuration, although formally they are defined as regular. We define an optimal-regular configuration as such that any subset of m actuators spans an m-dimensional velocity space. Since this densely constraints the work space, a more relaxed definition is needed. We therefore introduce the notion of k-singularity of a redundant mechanism which means that rigidifying k actuators will result in an optimal-regularity. We introduce an efficient algorithm to detect a k-singularity, give some examples for cases where m = 2, 3, and demonstrate our algorithm efficiency.

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