An Approach to Determining the Unknown Twist/Wrench Subspaces of Lower Mobility Serial Kinematic Chains

Abstract

Mainly drawing on screw theory and linear algebra, this paper presents an approach to determining the bases of three unknown twist and wrench subspaces of lower mobility serial kinematic chains, an essential step for kinematic and dynamic modeling of both serial and parallel manipulators. By taking the reciprocal product of a wrench on a twist as a linear functional, the underlying relationships among their subspaces are reviewed by means of the dual space and dual basis. Given the basis of a twist subspace of permissions, the causes of nonuniqueness in the bases of the other three subspaces are discussed in some depth. Driven by needs from engineering design, criteria, and a procedure are proposed that enable pragmatic, consistent bases of these subspaces to be determined in a meaningful, visualizable, and effective manner. Three typical examples are given to illustrate the entire process. Then, formulas are presented for the bases of the twist/wrench subspaces of a number of commonly used serial kinematic chains, which can readily be employed for the formulation of the generalized Jacobian of a variety of lower mobility parallel manipulators.

A simple and visually orientated approach for type synthesis of overconstrained 1T2R parallel mechanisms

“…6(a) as an example to explain this criterion. Utilizing the method developed in [32], the two linearly independent constraint wrenches, i.e.…”

Stiffness Modeling of Parallel Mechanisms at Limb and Joint/Link Levels

“…Leaving aside gravitational and inertial forces, the externally applied wrench w $ (expressed in ray-coordinates) imposed upon the platform is equilibrated by the sum of wrenches , collects the generalized reaction forces at the interface. The method for determining the unit wrenches of actuations and constraints of a serial kinematic chain can be found in [30]. Define, also, vector states that the instantaneous work delivered by the externally applied wrench w $ on the deflection twist t $ (expressed in axis-coordinates) of the platform equals the sum of the instantaneous work done by all ,…”

“…A reference frame  is attached to the base with O located at the centre of equilateral triangle A A A  , and the x , y and z axes parallel to x , y and z , respectively. Then, the unit wrenches in i W that is defined in (2) can be generated by the method given in [30]…”

Jacobian-based stiffness analysis method for parallel manipulators with non-redundant legs

“…Such set can be defined for any non-redundant leg, including those of lower mobility parallel manipulators. The unit twists of restriction are generally not unique, but Huang et al 32 describe how unique solutions can be found through the introduction of some simple additional conditions, which hold for most existing parallel manipulators.…”

“…The other vectors are illustrated in Figure 3, where b r i,l is the unit vector pointing along the lth link of leg i and b a i is the vector pointing from O 0 to the center of the third joint of leg i. (1) and (20) Next, taking the conditions posed by Huang et al 32 into account, a set of wrenches of constraint can also be identified…”