The Kinetostatic Optimization of Robotic Manipulators: The Inverse and the Direct Problems

Abstract

The design of a robotic manipulator begins with the dimensioning of its various links to meet performance specifications. However, a methodology for the determination of the manipulator architecture, i.e., the fundamental geometry of the links, regardless of their shapes, is still lacking. Attempts have been made to apply the classical paradigms of linkage synthesis for motion generation, as in the Burmester Theory. The problem with this approach is that it relies on a specific task, described in the form of a discrete set of end-effector poses, which kills the very purpose of using robots, namely, their adaptability to a family of tasks. Another approach relies on the minimization of a condition number of the Jacobian matrix over the architectural parameters and the posture variables of the manipulator. This approach is not trouble-free either, for the matrices involved can have entries that bear different units, the matrix singular values thus being of disparate dimensions, which prevents the evaluation of any version of the condition number. As a means to cope with dimensional inhomogeneity, the concept of characteristic length was put forth. However, this concept has been slow in finding acceptance within the robotics community, probably because it lacks a direct geometric interpretation. In this paper the concept is revisited and put forward from a different point of view. In this vein, the concept of homogeneous space is introduced in order to relieve the designer from the concept of characteristic length. Within this space the link lengths are obtained as ratios, their optimum values as well as those of all angles involved being obtained by minimizing a condition number of the dimensionally homogeneous Jacobian. Further, a comparison between the condition number based on the two-norm and that based on the Frobenius norm is provided, where it is shown that the use of the Frobenius norm is more suitable for design purposes. Formulation of the inverse problem—obtaining link lengths—and the direct problem—obtaining the characteristic length of a given manipulator—are described. Finally a geometric interpretation of the characteristic length is provided. The application of the concept to the design and kinetostatic performance evaluation of serial robots is illustrated with examples.

Structure Comparison and Optimal Design of 6-RUS Parallel Manipulator Based on Kinematic and Dynamic Performances

“…Another issue is that there are no direct relationships between the performance specifications and the structural parameters and solving the optimization problems will have several responses (Khan and Angeles, 2006).…”

“…Of course, the mentioned indices have some problems in dimension and physical interpretation (Khan and Angeles, 2006), because Jacobin matrix elements have similar dimension. Some methods are proposed to solve these problems.…”

“…First, optimizing the parallel manipulators is a multi-objective and multi-criterion problem based on different performance specifications and sometimes the mentioned specifications have diverse relationships. Another issue is that there are no direct relationships between the performance specifications and the structural parameters and solving the optimization problems will have several responses (Khan and Angeles, 2006).…”

“…First, optimizing the parallel manipulators is a multi-objective and multi-criterion problem based on different performance specifications and sometimes the mentioned specifications have diverse relationships. Another issue is that there are no direct relationships between the performance specifications and the structural parameters and solving the optimization problems will have several responses (Khan and Angeles, 2006).Hence, it is seen by studying the available references that in spite of great applications of 6-RUS parallel manipulators and proposed new various types by researches, structural difference between all manipulators and comparing their performances haven't been examined yet and no guidance has been proposed to select the proper dimensional parameters based on desired application. moreover, Although, a simplified type of 6-RUS manipulator in Liu et al(2002) has been examined based on kinematic dexterity index, no reference was seen to examine this manipulator based on dynamic index.…”

Position planning for collaborating robots and its application in neurosurgery

“…Here, the robot's characteristic length L is used as a normalization factor. The characteristic length is obtained by the procedure presented in [16]. Normalization of the Jacobian was made using the form shown in [17]:…”