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Technical Brief

Nonlinear, Phase-Based Oscillator to Generate and Assist Periodic Motions

[+] Author and Article Information
Juan De la Fuente

School of Engineering of Matter,
Transport and Energy,
Arizona State University,
Tempe, AZ 85281

Thomas G. Sugar, Sangram Redkar

The Polytechnic School,
Arizona State University,
Mesa, AZ 85212

Manuscript received October 7, 2016; final manuscript received February 17, 2017; published online March 9, 2017. Assoc. Editor: Hai-Jun Su.

J. Mechanisms Robotics 9(2), 024502 (Mar 09, 2017) (7 pages) Paper No: JMR-16-1296; doi: 10.1115/1.4036023 History: Received October 07, 2016; Revised February 17, 2017

Oscillatory behavior is important for tasks, such as walking and running. We are developing methods for wearable robotics to add energy to enhance or vary the oscillatory behavior based on the system's phase angle. We define a nonlinear oscillator using a forcing function based on the sine and cosine of the system's phase angle that can modulate the amplitude and frequency of oscillation. This method is based on the state of the system and does not use off-line trajectory planning. The behavior of a limit cycle is shown using the Poincaré–Bendixson criterion. Linear and rotational models are simulated using our phase controller. The method is implemented and tested to control a pendulum.

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Figures

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Fig. 1

Arbitrary point in the phase portrait of the system used to define the phase angle

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Fig. 2

Vector field and limit cycle of Eq. (8) for ωn = 1, ζ = 0.5, c = 1, and d = −1. The squares indicate the initial states. The diamonds indicate the final state. Both trajectories converge to the same limit cycle, one trajectory starts inside the limit cycle and another starts outside the limit cycle.

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Fig. 3

In case 1, the system oscillates at 2ωn with an amplitude of 0.08 m. c = 17.8885 and d = −120.

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Fig. 4

In case 2, the system oscillates at 0.5ωn with an amplitude of 0.12 m. c = 6.7082 and d = 45.

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Fig. 5

In case 3, the system response returns to a zero state. In the top graph, the output of the system is shown without a forcing function. In the bottom graph, the output of the system slows down using the phase-based forcing function with parameters c = −500 and d = −500. The forcing function forces the state of the system to zero in 1 order of magnitude faster for this case.

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Fig. 6

Case 4: Change of desired frequency at t = 10 s. The amplitude of the oscillations is maintained constant at A = 1 rad, and the frequency changes from ω = 5 rad/s to ω = 7 rad/s.

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Fig. 7

Case 5: Change of desired amplitude at t = 10 s. The frequency of the oscillations is maintained constant at ω = 7 rad/s, and the amplitude changes from Α = 1 rad to Α = 1.5 rad.

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Fig. 8

The top graph shows the position versus time. In the middle, velocity with Gaussian noise zero mean and variance four is shown. In the bottom, acceleration with Gaussian noise zero mean and variance four is shown. Regardless of the noise, the position output is smooth.

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Fig. 9

On the left, the experimental pendulum is shown. On the right, the block diagram of the system is shown.

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Fig. 10

Frequency transition. The plot shows the pendulum angular position θ for a change of values of c and d. The initial values are c1 = 1.849 and d1= −5.6744. The final values are c2 = 1.5411 and d2= −0.5521. The initial amplitude is A1 = 1.07 rad, and the final amplitude is the same at A2 = 1.06 rad. The initial frequency is ω1 = 6.04 rad/s, and the final frequency is ω2 = 4.79 rad/s.

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Fig. 11

Amplitude transition. The initial values are c1 = 1.84 and d1= −0.6625. The final values are c2 = 1.23 and d2= −0.4416. The initial amplitude is A1 = 1.31 rad, and the final amplitude is decreased, A2 = 0.69 rad. The initial frequency is ω1 = 4.8 rad/s, and the final frequency is ω2 = 4.9 rad/s.

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Fig. 12

The pendulum angular position θ versus time using the phase-based forcing function, c = 1.849 and d=−5.6744, and ω = 6 rad/s. An external force is introduced at t = 21 s and t = 40 s.

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