On the False Positives and False Negatives of the Jacobian Matrix in Kinematically Redundant Parallel Mechanisms

A robust geometric method of singularity avoidance for kinematically redundant planar parallel robot manipulators

“…In this section, an example mechanism is considered which is in a configuration where the Jacobian is singular but the rigidity matrix of the underlying graph of the robot manipulator is of full rank; indicating that the mechanism is in a non-singular configuration. A more comprehensive description of a similar example is given in [8].…”

“…Arguably the most common method is to compute the determinant of the Jacobian matrix of the robot manipulator, or to take the inverse of its condition number [7,2], where the robot manipulator is in a singularity if the result is equal to zero. However, it is known that these methods are not reliable when implemented on kinematically redundant parallel robot manipulators, more precisely on those which do not exhibit actuators connected in series, since the Jacobian has the potential to both incorrectly identify and fail to identify some singularities in the workspace [8].…”

“…Similarly, in [15] an overall motion planning scheme was developed, which aims to optimise the entire trajectory as a whole, that uses the same index as the previous reference. These methods are effective when applied to kinematically redundant robots with serially connected actuators, however they are limited when applied to architectures which exhibit no actuators connected in series, as it is for these architecutre where the Jacobian becomes unreliable as a means of singularity analysis [8].…”

Trailer-Maneuverability in N-Trailer Structures

“…determines the trailer-mobility ellipsoid in the task space. Upon the right-hand side of (17), and recalling that S (13) and (9)), one can write (omitting the arguments and introducing H := 1 j=N J j for a concise notation)…”

“…Since the first introduction of the fundamental manipulability analysis for serial robot manipulators in [3] (see also [4], [5]), the problems of motion capability and dexterity/agility have been investigated (in the fundamental meaning) for numerous mechanical and robotic systems, like: general constrained multibody systems [6], underactuated robots and devices [7], singlebody wheeled mobile robots [8], mobile manipulators [1], [9], multi-fingered devices [10], continuum robots [11], [12], redundant parallel mechanisms [13], or leader-follower networks [14]. Despite a common usage of the N-trailer structures, and in spite of numerous detailed study concerning their various properties (see, e.g., [15]- [22]), any generic trailer-maneuverability analysis has not been proposed for this kind of multi-body vehicles thus far (some analyses have been made for the tracks with trailers, as in [23], but they address a character/quality of motion in some particular scenarios rather than a generic capability of motion).…”