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Research Papers: Mechanisms and Robotics

# A Geometric Approach for Determining Inner and Exterior Boundaries of Workspaces of Planar Manipulators

[+] Author and Article Information
Dibakar Sen1

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, Indiadibakar@mecheng.iisc.ernet.in

B. Nagesh Singh

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, Indiabalajinagesh@rediffmail.com

These edges lie on segments connecting loops. The step is also active in some cases of partial rotation and some situation of degeneracy.

1

Corresponding author.

J. Mech. Des 130(2), 022306 (Jan 02, 2008) (9 pages) doi:10.1115/1.2821385 History: Received January 09, 2006; Revised April 02, 2007; Published January 02, 2008

## Abstract

In this paper, a method to determine the internal and external boundaries of planar workspaces, represented with an ordered set of points, is presented. The sequence of points are grouped and can be interpreted to form a sequence of curves. Three successive curves are used for determining the instantaneous center of rotation for the second one of them. The two extremal points on the curve with respect to the instantaneous center are recognized as singular points. The chronological ordering of these singular points is used to generate the two envelope curves, which are potentially intersecting. Methods have been presented in the paper for the determination of the workspace boundary from the envelope curves. Strategies to deal with the manipulators with joint limits and various degenerate situations have also been discussed. The computational steps being completely geometric, the method does not require the knowledge about the manipulator’s kinematics. Hence, it can be used for the workspace of arbitrary planar manipulators. A number of illustrative examples demonstrate the efficacy of the proposed method.

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## Figures

Figure 6

Boundary with segments not in the envelopes due to restricted rotatability. (a) Workspace and centroids (+), (b) open envelope curves, and (c) boundaries.

Figure 1

Representations of workspace with sample points for boundary determination. (a) Grid representation and (b) ordered points for sweep generation.

Figure 2

Instantaneous centers and critical points. (a) Center from two positions and (b) center from three centroids.

Figure 3

Generation of sweep envelopes

Figure 4

Intersecting envelopes decomposed into loops. Legend:- thick line: extracted segments; thin line: available segments; gray line: used segments. (a) First loop, (b) second loop, (c) third loop, and (d) all loops.

Figure 5

Loop and generators intersect for internal singularities. (a) Generators, centroids (+) and envelops, (b) loop and generator interaction, and (c) intersection of loop and generators.

Figure 7

Boundary with segments not in the envelopes due to nonsmooth sweep. (a) Workspace and centroids (+), (b) envelope curves, and (c) boundaries.

Figure 8

Determination of the right and left end segments for simple and self-intersecting end curves

Figure 9

Closing open envelopes with end segments. (a) Selecting the end segment and (b) attaching left end segment to envelopes.

Figure 10

Open end curves with end points not coinciding with extreme points

Figure 11

Detection and resolution of cusp singularity

Figure 12

Closed-loop manipulators (CLMs) used for data generation. (a) Four-bar CLM with pendant link, (b) revolving four bar CLM, and (c) five-bar CLM.

Figure 13

Workspace of a dyad with different sample densities. (a) First link longer, (b) boundary curves, (c) second link longer, and (d) boundary curves.

Figure 14

Workspaces of manipulators of the type in Fig. 1. (a) Workspace, (b) envelopes, (c) no hole, (d) workspace, (e) envelope, and (f) small hole.

Figure 15

Workspace generated through sweeping of noncircular curves from a manipulator as in Fig. 1. (a) Workspace, (b) boundaries, and (c) same at high resolution.

Figure 16

Workspace generated through sweeping of noncircular curves from Fig. 1. (a) Workspace, (b) boundaries, and (c) fine resolution of boundaries.

Figure 17

Dependence of computed boundary on the order of input scanning during data generation trimming operations are necessary. (a) Workspace, (b) scan order 1, and (c) scan order 2.

Figure 18

Workspace of manipulators with partial rotatability. (a) Workspace, (b) envelopes and end-segment, and (c) boundary segment.

Figure 19

Workspace boundary from complex envelopes. (a) Workspaces, (b) envelope curves, and (c) boundary curves.

Figure 20

Variation of computation time with the number of sample points in the workspace

## Errata

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