Research Papers

Design of Nonlinear Rotational Stiffness Using a Noncircular Pulley-Spring Mechanism

[+] Author and Article Information
Bongsu Kim

Department of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712
e-mail: bskim@utexas.edu

Ashish D. Deshpande

Assistant Professor
Department of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712
e-mail: ashish@austin.utexas.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received July 22, 2013; final manuscript received April 23, 2014; published online June 12, 2014. Assoc. Editor: David Dooner.

J. Mechanisms Robotics 6(4), 041009 (Jun 12, 2014) (9 pages) Paper No: JMR-13-1136; doi: 10.1115/1.4027513 History: Received July 22, 2013; Revised April 23, 2014

We present a new methodology for designing a nonlinear rotational spring with a desired passive torque profile by using a noncircular pulley-spring mechanism. A synthesis procedure for the shape of the noncircular pulley is presented. The method is based on an infinitesimal calculus approach that leads to an analytical solution, and the method is extended to address practical design issues related to the cable routing. Based on the synthesis method, an antagonistic spring configuration is designed for bilateral torque generation and is designed such that there is no slack in the routing cables. Two design examples are presented, namely, double exponential torque generation and gravity compensation for an inverted pendulum. Experiments with a mechanism for gravity compensation of an inverted pendulum validate our approach. We extend our approach to generate nonlinear torques at two joints by introducing the concept of torque decomposition. Experiments with a two-link robotic arm show that the gravitational forces from the masses on each link are accurately compensated for with our noncircular pulley-spring mechanisms.

Copyright © 2014 by ASME
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Fig. 1

A schematic diagram of the mechanism with a linear spring and noncircular pulley for counterbalancing an external desired torque τd (θ)

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Fig. 2

A schematic diagram showing two positions of line RP¯ represented by line “a” and “b” for two link position at θ and (θ + δθ), respectively. The pulley shape is synthesized by first determining the coordinate of the intersection point I.

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Fig. 3

An example of a pulley profile generated by determining a series of intersection points for the line of action of the spring force

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Fig. 4

Two cases of cable routing: cable runs on (a) or under the pulley (b). The geometrical modifications need to accommodate the cable thickness and radius of the routing pulley differently for the two cases.

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Fig. 5

Antagonistic spring configuration: (a) slack occurs when the antagonistic springs are attached without pretension and (b) there is no slack issue with pretension in the springs but concurrent synthesis on both sides of a pulley is required

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Fig. 6

The function of the desired torque profile (a) is split into two subfunctions (b) that are symmetrically transposed in order to have two independent one-sided pulley syntheses

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Fig. 7

Calculation of torque generation in a noncircular pulley-spring mechanism

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Fig. 8

An inverted pendulum model (a) and torque profile (the solid line) for gravity compensation of the pendulum and split torque (the dotted lines) to achieve antagonistic configuration (b)

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Fig. 9

A synthesized pulley shape for gravity compensation of an inverted pendulum

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Fig. 10

A torque profile (a) of humanlike joint stiffness and subprofiles (b) for synthesizing an antagonistic configuration

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Fig. 11

A synthesized pulley shape for humanlike double exponential joint stiffness

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Fig. 12

The antagonistic spring mechanism for compensating gravity force in an inverted pendulum. The figure (a) shows various parts of the mechanism and the figure (b) shows that the antagonistic spring-pulley statically balances the full weight of the pendulum in various positions.

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Fig. 13

Measured torque in the antagonistic spring mechanism versus theoretical required torque for compensating the circular-shaped weight at top of the pendulum

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Fig. 14

Required joint torques to balance the weight of an exoskeleton and human arm in a 2-DOF upper and forearm system

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Fig. 15

Spring configurations for a 2-DOF robotic arm: (a) a mono-articular spring on each joint, (b) a bi-articular spring, (c) an example of a bi-articular spring with circular pulleys; the left pulley is grounded and the right pulley is fixed to the last link (d) another example of a bi-articular spring with a parallelogram

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Fig. 16

A schematic diagram of a combination of noncircular pulleys and parallelogram for gravity compensation in a 2-DOF linkage

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Fig. 17

Demonstration of the passive gravity compensation in a 2-DOF planar mechanism; 1.55 kgf and 1.27 kgf weights are separately attached to the upper arm and the forearm linkage. The mechanism balances the weights at various configurations. At an extreme downward pose such as the configuration in (e), it was necessary to add a little more weight for balancing.

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Fig. 18

Joint angles at which the mechanism statically balances against fixed weights. It statically balanced with 1.55 kgf and 1.27 kgf at most of the regions represented by a circular dot. Heavier sets of weights were attached for the large angle values of the upper arm link.




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