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Research Papers

# Position and Force Analysis of a Planar Tensegrity-Based Compliant Mechanism

[+] Author and Article Information
Youngjin Moon

Department of Mechanical and Aerospace Engineering, Center for Intelligent Machines and Robotics,  University of Florida, Gainesville, FL 32611youngjin.moon@gmail.com

Carl D. Crane1

Department of Mechanical and Aerospace Engineering, Center for Intelligent Machines and Robotics,  University of Florida, Gainesville, FL 32611carl.crane@gmail.com

Rodney G. Roberts

Department of Electrical and Computer Engineering, FAMU-FSU College of Engineering,  Florida State University, Tallahassee, FL 32306rroberts@eng.fsu.edu

ri ’s are not shown in Fig. 1 because these distance vectors are defined by a reference coordinate system and its origin.

The planar unitized screw, $is generally defined as 3 × 1 vector of [S T , r × S ]T in screw theory, where S is a direction and r is a distance vector from the origin of the reference to any point on the line. Therefore,$ = [cosθ, sinθ, rx sin θ − ry cos θ]T , where rx and ry are x and y direction components of r for each leg connector.

This planar stiffness matrix is partially symmetric due to the last term of Eq. 9. That is to say, the upper-left 2 × 2 partial matrix of the matrix is symmetric.

Generally, all possible combinations of J ′ are nonsingular when the matrix, [$1 ,$2 ,$3 ,$4 ] has a full rank. When it fails, a polynomial system, built from geometric constraints, static equilibrium, and the potential energy in the springs, is underdetermined system, then numerical methods are required to solve the system. The failure cases happen when all four lines passing through the leg connectors are parallel or concurrent. Those cases are rare, and they can be avoided by a specific joint configuration.

It is very difficult to find the conditions for singularity and deduce its geometrical meaning because |A K | comes from an 8 × 8 matrix and, thus, forms an extremely complicated equation.

1

Corresponding author.

J. Mechanisms Robotics 4(1), 011004 (Feb 03, 2012) (8 pages) doi:10.1115/1.4005531 History: Received September 14, 2010; Accepted September 09, 2011; Published February 03, 2012; Online February 03, 2012

## Abstract

This paper presents an analysis of a planar tensegrity-based mechanism. In this study, a moving body is joined to ground by four compliant leg connectors. Each leg connector is comprised of a spring in series with an adjustable length piston. Two problems are solved in this paper. In the first, the values of the four spring constants and free lengths are given and the lengths of the four pistons are determined such that (1) the top body is positioned and oriented at a desired pose, (2) the top body is at equilibrium while a specified external wrench is applied, and (3) the total potential energy stored in the four springs equals some desired value. In the second problem, the values for the four spring free lengths are given and the values for the four spring constants and the lengths of the four pistons are determined such that conditions (1) and (2) from above are met and also the instantaneous stiffness matrix of the top body equals a specified set of matrix values. The paper formulates the solutions of these two problems. Numerical examples are presented.

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## Figures

Figure 6

Example 3: Piston lengths (solution A)

Figure 7

Example 3: Piston length (solution B)

Figure 8

Example 3: Pose change (solution A)

Figure 9

Example 3: Pose change (solution B)

Figure 1

Mechanism schematic

Figure 2

Compliant leg connector

Figure 3

Example 1: Force magnitudes for change in magnitude of the external wrench

Figure 4

Example 2: Pose of the mechanism

Figure 5

Example 3: Path trajectory

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