A Constraint-Based Approach to Manipulator Kinematics and Singularities

Abstract

In this thesis, a differential-geometric approach to the kinematics of multibody mechanisms is introduced that enables analysis of singularities of both serial and parallel manipulators in a flexible and complete way. Existing approaches such as those of Gosselin and Angeles [1], Zlatanov et al. [2] and Park and Kim [3] make use of a combination of joint freedoms and constraints and so build in assumptions. In contrast, this new approach is solely constraint-based, avoiding some of the shortcomings of these earlier theories.

The proposed representation has two core ingredients. First, it avoids direct reference to the choice of inputs and their associated joint freedoms and instead focuses on a kinematic constraint map (KCM), defined by the constraints imposed by all joints and not requiring consideration of closure conditions arising from closed loops in the design. The KCM is expressed in terms of pose (i.e. position and orientation) variables, which are the coordinates of all the manipulator’s links with respect to a reference frame. The kinematics of a given manipulator can be described by means of this representation, locally and globally. Also, for a family of manipulators defined by a specific architecture, the KCM will tell us how the choice of design parameters (e.g. link lengths) affects these kinematic properties within the family.

At a global level, the KCM determines a subset in the space of all pose variables, known as the configuration space (C-space) of the manipulator, whose topology may vary across the set of design parameters. The Jacobian (matrix of first-order partial derivatives) of the KCM may become singular at some specific choices of pose variables. These conditions express a subset called the singular set of the C-space. It is shown that if a family of manipulators, parametrised by a manifold Rd of design parameters, is “well-behaved” then the pose variables can be eliminated from the KCM equations together with the conditions for singularities, to give conditions in terms of design parameters, that define a hypersurface in Rd of manipulators in the class that exhibit C-space singularities. These are referred to as Grashof-type conditions, as they generalise classically known inequalities classifying planar 4-bar mechanisms due to Grashof [4].

Secondly, we develop the theory to incorporate actuator space (A-space) and workspace (W-space), based on a choice of actuated joints or inputs and on the manipulator’s end-effector workspace or outputs. This will facilitate us with a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator’s C-space. This provides new insight into the structure of the forward and inverse kinematics, especially for parallel manipulators.

The theory is illustrated by a number of applications, some of which recapitulate classical or known results and some of which are new.