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TECHNICAL PAPERS

# Path Tracking of Parallel Manipulators in the Presence of Force Singularity

[+] Author and Article Information
C. K. Kevin Jui, Qiao Sun

Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4

While the transformed parallelepiped becomes degenerate at singularity, a facet only becomes degenerate when null $(A)$ falls into the $(n−1)$ basis of the corresponding facet in the joint space. We limit our analysis to cases where $ac$ only passes through non-degenerate facets, which are the typical situations encountered in reality.

If abs $(det([ac−Ti])) where $el$ is a small positive real number to be specified, the $ith$ pair of facets are considered either degenerate or being parallel to $ac$.

At singularity, $s̈min≠s̈max$ iff $ac∊range(A)$ and the line directed by $ac$ passes through the transformed parallelepiped, which is already degenerate.

J. Dyn. Sys., Meas., Control 127(4), 550-563 (Jan 17, 2005) (14 pages) doi:10.1115/1.2098893 History: Received December 03, 2003; Revised January 17, 2005

## Abstract

Parallel manipulators are uncontrollable at force singularities due to the infeasibly high actuator forces required. Existing remedies include the application of actuation redundancy and motion planning for singularity avoidance. While actuation redundancy increases cost and design complexity, singularity avoidance reduces the effective workspace of a parallel manipulator. This article presents a path tracking type of approach to operate parallel manipulators when passing through force singularities. We study motion feasibility in the neighborhood of singularity and conclude that a parallel manipulator may track a path through singular poses if its velocity and acceleration are properly constrained. Techniques for path verification and tracking are presented, and an inverse dynamics algorithm that takes actuator bounds into account is examined. Simulation results for a planar parallel manipulator are given to demonstrate the details of this approach.

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## Figures

Figure 8

Phase portrait

Figure 9

Types of upper boundary curves

Figure 10

Types of lower boundary curves

Figure 20

Path tracking along path 3

Figure 21

Path tracking along path 5

Figure 6

Minimum and maximum acceleration—Algorithm I

Figure 7

Feasible and unfeasible trajectory slope

Figure 11

Lower boundary curves of path 2

Figure 1

Planar two DOF parallel manipulator

Figure 2

Figure 3

Parametrized paths

Figure 4

Bounded region of τ in the joint space (a), and the transformed region in task space (b)

Figure 5

Facets of the bounded region of τ (a), and facets of the transformed region (b)

Figure 12

Lower boundary curves of path 3

Figure 13

UVC (thick solid line) and LVC (thin solid line) of the six paths illustrated in Fig. 3

Figure 14

Boundary curves: (a) path 1—untrackable; (b) path 2—trackable; (c) path 3—trackable; (d) path 5—trackable

Figure 15

Example trajectories: (a) path 2; (b) path 3; (c) path 5; (d) commonly used trajectories

Figure 16

Actuator forces required for trajectory tracking

Figure 17

Conventional control method

Figure 18

Modified control method

Figure 19

Path tracking along path 2

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