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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Revzen, S. , and Guckenheimer, J. M. , 2008, “ Estimating the Phase of Synchronized Oscillators,” Phys. Rev., 78(5), p. 051907.
Tilton, A. K. , Hsiao-Wecksler, E. T. , and Mehta, P. G. , 2012, “ Filtering With Rhythms: Application to Estimation of Gait Cycle,” American Control Conference (ACC), Montréal, Canada, June 27–29, pp. 3433–3438.
Kerestes, J. , Sugar, T. G. , Flaven, T. , and Holgate, M. , 2014, “ A Method to Add Energy to Running Gait: PogoSuit,” ASME Paper No. DETC2014-34406.
Kerestes, J. , Sugar, T. G. , and Holgate, M. , 2014, “ Adding and Subtracting Energy to Body Motion: Phase Oscillator,” ASME Paper No. DETC2014-34405.
Sugar, T. G. , Bates, A. , Holgate, M. , Kerestes, J. , Mignolet, M. , New, P. , Ramachandran, R. K. , Redkar, S. , and Wheeler, C. , 2015, “ Limit Cycles to Enhance Human Performance Based on Phase Oscillators,” ASME J. Mech. Rob., 7(1), p. 011001. [CrossRef]
New, P. , Wheeler, C. , and Sugar, T. G. , 2014, “ Robotic Hopper Using Phase Oscillator Controller,” ASME Paper No. DETC2014-34188.
Ronsse, R. , Vitiello, N. , Lenzi, T. , van den Kieboom, J. , Chiara Carrozza, M. , and Jan Ijspeert, A. , 2010, “ Adaptive Oscillators With Human-in-the-Loop: Proof of Concept for Assistance and Rehabilitation,” 3rd IEEE RAS and EMBS International Conference on Biomedical Robotics and Biomechatronics (BioRob), Tokyo, Japan, Sept. 26–29, pp. 668–674.
Rinderknecht, M. D. , Delaloye, F. A. , Crespi, A. , Ronsse, R. , and Jan Ijspeert, A. , 2011, “ Assistance Using Adaptive Oscillators: Robustness to Errors in the Identification of the Limb Parameters,” IEEE International Conference on Rehabilitation Robotics (ICORR), Zurich, Switzerland, June 29–July 1.
Righetti, L. , Buchli, J. , and Jan Ijspeert, A. , 2009, “ Adaptive Frequency Oscillators and Applications,” Open Cybern. Syst. J., 3(1), pp. 64–69. [CrossRef]
Righetti, L. , Buchli, J. , and Ijspeert, A. J. , 2006, “ Dynamic Hebbian Learning in Adaptive Frequency Oscillators,” Physica D.: Nonlinear Phenomena, 216(2), pp. 269–281. [CrossRef]
Seo, K. , Hyung, S. Y. , Choi, B. K. , Lee, Y. , and Shim, Y. , 2015, “ A New Adaptive Frequency Oscillator for Gait Assistance,” IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, May 26–30, pp. 5565–5571.
Rouse, E. J. , Gregg, R. D. , Hargrove, L. J. , and Sensinger, J. W. , 2013, “ The Difference Between Stiffness and Quasi-Stiffness in the Context of Biomechanical Modeling,” IEEE Trans. Biomed. Eng., 60(2), pp. 562–568. [CrossRef] [PubMed]
Asano, F. , Yamakita, M. , Kamamichi, N. , and Luo, Z. W. , 2004, “ A Novel Gait Generation for Biped Walking Robots Based on Mechanical Energy Constraint,” IEEE Trans. Rob. Autom., 20(3), pp. 565–573. [CrossRef]
Gregg, R. D. , and Sensinger, J. W. , 2014, “ Towards Biomimetic Virtual Constraint Control of a Powered Prosthetic Leg,” IEEE Trans. Control Syst. Technol., 22(1), pp. 246–254. [CrossRef] [PubMed]
Gregg, R. D. , Lenzi, T. , Fey, N. P. , Hargrove, L. J. , and Sensinger, J. W. , 2013, “ Experimental Effective Shape Control of a Powered Transfemoral Prosthesis,” IEEE International Conference on Rehabilitation Robotics (ICORR), Seattle, WA, June 24–26, pp. 1–7.
Linkens, D. A. , 1977, “ The Stability of Entrainment Conditions for RLC Coupled Van der Pol Oscillators Used as a Model for Intestinal Electrical Rhythms,” Bull. Math. Biol., 39(3), pp. 359–372. [PubMed]
Matsuoka, K. , 1987, “ Mechanisms of Frequency and Pattern Control in the Neural Rhythm Generators,” Biol. Cybern., 56(5–6), pp. 345–353. [CrossRef] [PubMed]
Kuramoto, Y. , 1975, Self-Entrainment of a Population of Coupled Non-Linear Oscillators (Lecture Notes in Physics), Springer, Berlin, pp. 420–422.
De la Fuente Valadez, J. O. , 2016, “ Nonlinear Phase Based Control to Generate and Assist Oscillatory Motion With Wearable Robotics,” Ph.D. dissertation, Arizona State University, Mesa, AZ.
Khalil, H. K. , 2002, Nonlinear Systems, 3rd ed., Prentice Hall, Upper Saddle River, NJ.


Grahic Jump Location
Fig. 1

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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
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.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In