Singularity-Consistent Parameterization of Robot Motion and Control

Abstract

The inverse kinematics problem is formulated as a parameterized autonomous dynamical system problem, and respective analysis is carried out. It is shown that a singular point of work space can be mapped either as a critical or a noncritical point of the autonomous system, depending on the direction of approach to the singular point. Making use of the noncritical mapping, a closed-loop kinematic controller with asymptotic stability and velocity limits along degenerate singular or near-singular paths is designed. The authors introduce a specific type of motion along the reference path, the so-called natural motion. This type of motion is obtained in a straightforward manner from the autonomous dynamical system and always satisfies the motion constraint at a singular point. In the vicinity of the singular point, natural motion slows down the end-effector speed and keeps the joint velocity bounded. Thus, no special trajectory replanning will be required. In addition, the singular manifold can be crossed, if necessary. Further on, it is shown that natural motion constitutes an integrable motion component. The remaining, nonintegrable motion component is shown to be helpful in solving a problem related to the critical point mapping of the autonomous system. The authors design a singularity-consistent resolved acceleration controller, which they then apply to singular or near-singular trajectory tracking under torque limits. Finally, the authors compare the main features of the singularity-consistent method and the damped-least-squares method. It is shown that both methods introduce a so-called algorithmic error in the vicinity of a singular point. The direction of this error is, however, different in each method. This is shown to play an important role for system stability.

Trajectory tracking with a 6R serial industrial robot with ordinary and non-ordinary singularities

“…Moreover codimension one singularities (i.e., J has a maximum rank deficiency of one) will be considered because singularities with a higher order codimension are not very common in practice. However with a recursive singular-value-decomposition(SVD) the codimension d problem is reduced to codimension one [5], although the computational time is increased.…”

“…In fact, this approach is equivalent to the adjoint Jacobian method: the direction and magnitude of the motion are decoupled; no direction error is introduced; a subsequent time analysis based on parameter η must be considered in order to achieve the desired global time. If the parameter η is constant and determined from the initial kinetic energy (i.e., with an initial nonzero differential motion) it is obtained a differential motion called natural [5].…”

“…If singular configurations are simply avoided the manipulator's workspace is reduced, certain types of operations are impossible (e.g., full-arm extension using straight-line motion), and the motion-planning process is complicated. However, works on the singularity problem indicate that with proper time scaling [1] or path reparameterization [2][3][4][5] it is possible to follow any path near and at a singularity, without incurring large joint velocities. Therefore, the aim of this research is to provide a unified view of the singularity problem and to develop a trajectory control valid not only for ordinary singularities, but also for nonordinary singularities.…”

An Adaptive Control for a Free-Floating Space Robot by Using Inverted Chain Approach

“…Note that the control law (9) can be achieved under the condition when H em is nonsingular. Since several researches have already been proposed the treatment of the singularity problem [7]- [9], it is out of focus in this paper.…”

An adaptive control for a free-floating space robot by using inverted chain approach

“…Note that the control law (9) can be achieved under the condition when is nonsingular. Since several researches have already been proposed the treatment of the singularity problem [Nenchev et al, 2000,Tsumaki et al, 2001, Senft & Hirzinger, 1995, Nakamura & Hanafusa, 1986, it is out of focus in this chapter.…”