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Research Papers

Synthesis of Point Planar Elastic Behaviors Using Three-Joint Serial Mechanisms of Specified Construction

[+] Author and Article Information
Shuguang Huang

Department of Mechanical Engineering,
Marquette University,
Milwaukee, WI 53201-1881
e-mail: huangs@marquette.edu

Joseph M. Schimmels

Department of Mechanical Engineering,
Marquette University,
Milwaukee, WI 53201-1881
e-mail: j.schimmels@marquette.edu

1Corresponding author.

Manuscript received April 28, 2016; final manuscript received November 1, 2016; published online December 2, 2016. Assoc. Editor: Yuefa Fang.

J. Mechanisms Robotics 9(1), 011005 (Dec 02, 2016) (11 pages) Paper No: JMR-16-1122; doi: 10.1115/1.4035189 History: Received April 28, 2016; Revised November 01, 2016

This paper presents methods for the realization of 2 × 2 translational compliance matrices using serial mechanisms having three joints, each either revolute or prismatic and each with selectable compliance. The geometry of the mechanism and the location of the compliance frame relative to the mechanism base are each arbitrary but specified. Necessary and sufficient conditions for the realization of a given compliance with a given mechanism are obtained. We show that, for an appropriately constructed serial mechanism having at least one revolute joint, any single 2 × 2 compliance matrix can be realized by properly choosing the joint compliances and the mechanism configuration. For each type of three-joint combination, requirements on the redundant mechanism geometry are identified for the realization of every point planar elastic behavior at a given location, just by changing the mechanism configuration and the joint compliances.

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Figures

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Fig. 1

One type of serial compliant mechanism (RPR) with variable stiffness actuators

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Fig. 2

Joint twists in a serial mechanism. (a) For a prismatic joint, the joint twist ti is a unit vector along the prismatic joint axis. It is independent of the location of the joint. (b) For a revolute joint, the joint twist ti is orthogonal to the position vector ri and depends on the location of the joint.

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Fig. 3

Realization of a given compliance with a two-joint mechanism having a specified configuration. Joint twist t2 of J2 must be collinear with line l1⊥Kt1 at point O to realize the given C.

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Fig. 4

Relationship between joint twists ti and lines li. (a) If two joints are specified, the acceptable area for t3 is determined by lines l1 and l2. (b) Realization condition: t1 must be between and adjacent to lines l2 and l3; t2 must be between and adjacent to lines l1 and l3; and t3 must be between and adjacent to lines l1 and l2.

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Fig. 5

Geometric parameters of a PRR mechanism. The locus of J2 locations is line l, and the locus of J3 locations is circle C.

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Fig. 6

Extreme positions of a PRR mechanism. The extreme positions of link-3 occur when link-2 is parallel to line OO′. (a) Link-1 and link-2 are collinear, and J3 is at the intersection of line l1 and circle C. (b) Link-2 folds over link-1, and J3 is at the intersection of line l2 and circle C.

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Fig. 7

Geometric parameters of an RPR mechanism. L1 is the perpendicular distance from revolute joint J1 to the axis along which joint J3 is guided.

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Fig. 8

Extreme positions of an RPR mechanism. (a) The extreme position of link-1 occurs when J1A is parallel to link-3 (OJ3). (b) The extreme position of link-3 occurs when J3 is at the intersection point of circles C1 and C2. Link-3 cannot enter the interior of the shaded area.

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Fig. 9

Geometric parameters of an RRP mechanism. L3 is the (perpendicular) distance from the mechanism endpoint O to the prismatic axis.

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Fig. 10

Extreme positions of an RRP mechanism. The extreme position of link-3 occurs when prismatic joint J3 reaches circles C1 and C2. When J3 is on circle C1, link-1 and link-2 are fully extended. When J3 is on circle C2, link-2 folds over link-1.

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Fig. 11

Extreme positions of an RRP mechanism. (a) If θ3M does not exist, link-3 can rotate through θ3=180 deg in going from position θ3m to position −θ3m. (b) If θ3m does not exist, link-3 can rotate through θ3=0 in going from position θ3M to position −θ3M. For each case, prismatic joint J3 will pass through point A of link AB.

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Fig. 12

Geometric parameters of a PPR mechanism. The motion of link-3 is restricted by the limits of the two prismatic joints.

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Fig. 13

Geometric parameters of a PRP mechanism. The extreme positions can be determined by the limits of J1, L1M, and L1m.

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Fig. 14

An RPP mechanism. Link-1 and link-3 have the same range of rotation variation.

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Fig. 15

(a) A specified RRP mechanism having given link length L1=3. The position of joint base J1 relative to the compliance frame origin O is specified L0=2. (b) Configurations of the mechanism: Synthesis of C with two compliant joints J1 and J2 or J1 and J3.

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Fig. 17

Extreme positions of a PRP mechanism. The minimum angle of link-3 is determined by the upper limit of link-1, L1M.

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