A Sequential Method for the Singularity Free Design of a Planar 3-arm Parallel Robot 1

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Yawei Yang , John F. O’Brien

Department of Electrical and Computer Engineering University of Wyoming 1000E Univ. Ave. Laramie, WY, 82070

Introduction

Singular line and special singularities

• A parallel robot is a device for performing manipulations, where the end effector is connected to the base via multiple kinematic chains. Any two chains thus form a closed loop. • Comparation of the singularities between parallel and serial robots. Because the parallel robots usually contain passive joints, their singular conditions are different from serial robots. They contain unstable singularities which will not happen to serial robots.

Figure 1. A typical serial robot

Figure 2. UW hexapod parallel robot

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Figure 6. Singularity line for joint 3 3 6 E 4

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Figure 7. Special singularity • For planar 3-RPR mechanism, – the unstable singularities happen when the three arms intersect at a point including infinity. – The unmanipulable singularities happen when an arm has zero length. For the first condition, ∃α1, α2, α3

Singularities of parallel robots 3 6

• Unmanipulable singularities – Configurations where a DOF is lost – Happens to both serial and parallel mechanisms.

• Consider the first two arms, for each pose inside the workspace, they intersect at a point T . If all the three arms intersect at T , the third base joint must be located on the line passes both 6 and T . We call the line L6T the singular line. • For some configurations the singular line becomes arbitrary, we call them special singularities (As shown in figure 7). In order to avoid them, we should modify the design of the first two arms.

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– The workspace does not contain the poses with ~p14 and ~p25 intersect at joint 6. – Both arms are not be zero length. • 3: General design rules for planar 3-RPR parallel mechanism ~p23·(~p25×~z) p14×~z) and f = be the left and right side of the second Let f1 = ~~pp13·(~ 2 ·(~ p ×~ z) ~ p ·(~ p ×~ z) 14 46 56 25 equation in (4). Using the properties of these functions we can identify if there exists a location for the third joint to get a singularity free workspace. Rules are derived to improve the existence of the solution set for the location of joint 3. • 4: Fiding the solution set for joint 3 For a fixed joint 3, f1, f2 are monotonic functions on any continuous domain. We can design the location of joint 3 such that f1 max < f2 min or f2 max < f1 min. For example, if for the entire workspace, f1, f2 are bounded, then the extrema are at the borders of their continuous domains.

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Examples for Planar 3-RPR Parallel Robot 10

~p1 + α1~p14 = ~p2 + α2~p25 ~p1 + α1~p14 = ~p3 + α3~p36 ~p3 + α3~p36 = ~p2 + α2~p25

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Figure 3. Unmanipulable singularity 3

• Unstable singularities – The end-effector can not resist certain applied task wrenches. – Unique to parallel mechanisms.

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From these equations we derive a useful description of the unstable singular conditions of the 3-RPR mechanism from a design perspective.

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~p23 · (~p36 ×~z) ~p12 · (~p25 ×~z) = ~p56 · (~p36 ×~z) ~p45 · (~p25 ×~z) ~p23 · (~p25 ×~z) ~p13 · (~p14 ×~z) = ~p56 · (~p25 ×~z) ~p46 · (~p14 ×~z) ~p12 · (~p14 ×~z) ~p13 · (~p36 ×~z) = ~p45 · (~p14 ×~z) ~p46 · (~p36 ×~z)

Figure 4. Unstable singularity

Design Method 3

The design process is segregated into three phases:

• calculate the solution set for the object parameters.

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y {E} x E 4 {o}y

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• Using the design method presented in this poster, we can find a solution set for 2DOF and 3DOF planar 3-RPR singularity free workspace design. Two specific example depicted in Figure 7 and 8.

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Figure 8. Solution set for the location of joint 3 of 2DOF 20

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Figure 9. Solution set for the location of joint 3 of 3DOF Acknowledgements: This work was funded in part by a NASA Space Grant and the State of Wyoming.

Design Rules

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These conditions are simultaneously satisfied at a singular pose.

• choose the parameters for one of the three arms as the design variables. • modify the parameters of the other two arms to improve the existence of singularity free workspace.

(1) (2) (3)

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Figure 5. A typical planar 3-RPR mechanism

• 1: Description of singularity free workspace design for planar 3-RPR parallel robot If one of the conditions described by equations in (4) is not satisfied for the whole workspace and the workspace does not contain unmanipulable singularities, then mechanism is singularity free (including special singularities). • 2: Design rules to avoid special singularities for planar 3-RPR parallel robot – The workspace does not contain the poses with ~p14 k ~p45 k ~p25.

References [1] J. -P. Merlet, Parallel Robots (Series: Solid Mechanics and Its Applications , Vol. 128). Dordrecht, The Netherlands: Kluwer Academic Publishers, 2000. [2] J. T. Wen and J. F. O’Brien, “Determination of Unstable Singularities in Parallel Robots,” IEEE Trans. on Robot. and Automat., vol. 19, no. 4, pp 720-725, August 2003. [3] J. F. O’Brien and J. T. Wen, “On Kinematic Instability of Parallel Robots,” Electronic Journal of Computational Kinematics, 1(1), February 2002 [4] http://www.parallemic.org [5] http://www-sop.inria.fr/coprin/equipe/merlet/merlet eng.html