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Research Papers

A High-Fidelity Harmonic Drive Model

[+] Author and Article Information
Curt Preissner1

 The Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439 preissner@aps.anl.gov

Thomas J. Royston

UIC Department of Mechanical and Industrial Engineering, 842 West Taylor Street, MC 251, Chicago, IL 60607 troyston@uic.edu

Deming Shu

 The Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439 shu@aps.anl.gov

1

Corresponding author.

J. Dyn. Sys., Meas., Control 134(1), 011002 (Dec 02, 2011) (13 pages) doi:10.1115/1.4005041 History: Received October 12, 2009; Revised July 07, 2011; Published December 02, 2011; Online December 02, 2011

In this paper, a new model of the harmonic drive transmission is presented. The purpose of this work is to better understand the transmission hysteresis behavior while constructing a new type of comprehensive harmonic drive model. The four dominant aspects of harmonic drive behavior—nonlinear viscous friction, nonlinear stiffness, hysteresis, and kinematic error—are all included in the model. The harmonic drive is taken to be a black box, and a dynamometer is used to observe the input/output relations of the transmission. This phenomenological approach does not require any specific knowledge of the internal kinematics. In a novel application, the Maxwell resistive-capacitor hysteresis model is applied to the harmonic drive. In this model, sets of linear stiffness elements in series with Coulomb friction elements are arranged in parallel to capture the hysteresis behavior of the transmission. The causal hysteresis model is combined with nonlinear viscous friction and spectral kinematic error models to accurately represent the harmonic drive behavior. Empirical measurements are presented to quantify all four aspects of the transmission behavior. These measurements motivate the formulation of the complete model. Simulation results are then compared to additional measurements of the harmonic drive performance.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

An exploded view of a harmonic drive showing the three components

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Figure 2

Schematics of the harmonic drive dynamometer and fixture: (a) The dynamometer configured to apply the input to the wave generator while the flexspline is fixed; (b) The top and side view of the fixture used to apply an input to the flexspline while the wave generator is fixed; and (c) the dynamometer as configured for measuring the kinematic error, viscous dissipation, and dynamic response of the HDT/dynamometer system

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Figure 3

These are examples of the input histories used for hysteresis identification: (a) monotonically decreasing series of torque input extrema, (b) sequence of alternating local maxima and minima torque inputs, and (c) variation between the same input torque extremes

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Figure 4

Hysteresis relations for the harmonic drive: (a) experiment 1: torque versus torsion—torque was controlled at the wave generator while the flexspline was fixed; (b) experiment 2: torsion versus torque—torsion was controlled at wave generator, flexspline torque measured while the flexspline was fixed; (c) experiment 3: torsion versus torque—torsion was controlled at the flexspline while the wave generator was fixed

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Figure 5

Input torque as a function of the average wave generator angular velocity. The circles are measured data and the line is the arctangent function fit to the data.

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Figure 6

The kinematic error, with respect to flexspline revolutions, measured for a constant wave generator velocity. In the ideal case, the flexspline position would track the wave generator input within the factor of the gear ratio.

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Figure 7

Harmonic drive time domain and frequency response as a function of a 5-Hz sinusoidal excitation: (a) the time domain data from the experiment and (b) the FRF data from the experiment. These charts quantify the steady state dynamic response of the harmonic drive when coupled to a system of known inertia and stiffness.

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Figure 16

The Everett surface corresponding to the set of first-order reversal curves shown in Figure 4

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Figure 17

(a) A set of hysteresis loops for the parallel series MRC hysteresis model (66 elements) (b) Comparison of the model output to the measured response: Solid—measured transmission response, and dashed—simulated transmission response. The average difference between the empirical data and the simulation was 2.7% while the maximum difference was 6.4 Nm.

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Figure 18

Schematic of the two-element parallel-series HDT hysteresis model. TW is the torque applied to the flexspline and θTSN is the torsion between the wave generator and the flexspline. β1 and β2 are the spring stiffnesses. θb is the position of the slider and erc is the Coulomb friction force associated with the slider. The pivoting lever represents the sign reversal due to the HDT kinematics.

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Figure 19

(a) A set of hysteresis loops for the two-element parallel series MRC hysteresis model. (b) Comparison of the model output to the measured response: Solid—measured transmission response and dashed—simulated transmission response. The average difference between the empirical data and the simulation was 1.8%.

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Figure 20

The harmonic drive kinematic error: The upper chart shows the spectrum of displacement variation magnitudes as a function of cycles per flexspline rotation. The lower chart is the simulated kinematic error when it is approximated with the six largest components from the spectrum.

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Figure 21

Simulation time and frequency response as a function of the experimentally acquired input torque signal: (a) the time domain acceleration data of the test mass, and (b) the FRF data between the input torque and the test mass acceleration. These charts quantify the steady state dynamic response of the harmonic drive when coupled to a system of known inertia and stiffness.

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Figure 8

The basic kinematic representation of an HDT showing the three ports

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Figure 9

Schematic of the dynamometer, including the harmonic drive model (within the larger dashed box) with the velocity-dependent friction (β), kinematic error (circle), hysteresis effects (smaller dashed box), and lumped stiffnesses and inertias for the measurement system. The position of the circular spline θCS is fixed. The end condition can be changed between a free shaft, a fixed shaft, and a known inertia.

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Figure 10

(a) Schematic of the single parallel elastic-slide element HDT hysteresis model, and (b) the corresponding hysteresis output. TW is the wave generator torque and θTSN is the torsion between the wave generator and the flexspline. Y is the breakaway torque and α and χ are the last maxima and minima, respectively. The pivoting lever represents the sign reversal due to the HDT kinematics.

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Figure 11

To determine KT , the ascending and descending portions of the torque torsion curves shown in Fig. 4, were approximated with a linear least squares fit, represented by the dotted lines. The distance between the two lines, at the colored triangles, was calculated to find the breakaway torque Y.

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Figure 12

(a) A set of hysteresis loops for the single elastic-slide element hysteresis model. (b) Comparison of the model output to the measured response: Solid—measured transmission response, and dashed—simulated transmission response. The average difference between the empirical data and the simulation was 0.84%.

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Figure 13

Schematic of the parallel-series (MRC) HDT hysteresis model, consisting of n elastic-slide elements. θTSN is the model input and is the torsion between the wave generator and the flexspline. Ψ is the model output and is the hysteresis torque at the flexspline. βi is the spring stiffnesses. θb, i are the positions of the sliders, and ei is the Coulomb friction forces associated with each slider. The pivoting lever represents the sign reversal due to the HDT kinematics.

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Figure 14

Preisach hysteresis relay operator γ[u(t)]

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Figure 15

Functional relationships of the classical Preisach model: (a) the weighting function μ(x, y) and (b) the Everett function F(x, y) from first-order transition curves

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