In this paper we present a novel method to obtain the basic frequency of an unknown periodic signal with an arbitrary waveform, which can work online with no additional signal processing or logical operations. The method originates from non-linear dynamical systems for frequency extraction, which are based on adaptive frequency oscillators in a feedback loop. In previous work, we had developed a method that could extract separate frequency components by using several adaptive frequency oscillators in a loop, but that method required a logical algorithm to identify the basic frequency. The novel method presented here uses a Fourier series representation in the feedback loop combined with a single oscillator. In this way it can extract the frequency and the phase of an unknown periodic signal in real time and without any additional signal processing or preprocessing. The method determines the Fourier series coefficients and can be used for dynamic Fourier series implementation. The proposed method can be used for the control of rhythmic robotic tasks, where only the extraction of the basic frequency is crucial. For demonstration several highly non-linear and dynamic periodic robotic tasks are shown, including also a task where an electromyography (EMG) signal is used in a feedback loop.
AnCAtkesonCHollerbachJ (1988) Model-based Control of A Robot Manipulator. Cambridge, MA: The MIT Press.
2.
BaileyA (2004) Biomimetic Control with a Feedback Coupled Nonlinear Oscillator: Insect Experiments, Design Tools, and Hexapedal Robot Adaptation Results. PhD Thesis, Stanford University.
3.
BuchliJIjspeertA (2004) A simple, adaptive locomotion toy-system. In SchaalSIjspeertABillardAVijayakumarSHallamJMeyerJ (eds), From Animals to Animats 8. Proceedings of the Eighth International Conference on the Simulation of Adaptive Behavior (SAB’04). Cambridge, MA: The MIT Press, pp. 153–162.
4.
BuchliJRighettiLIjspeertAJ (2005) A dynamical systems approach to learning: A frequency-adaptive hopper robot. In CapcarrereMSFreitasAABentleyPJJohnsonCGTimmisJ (eds), Advances in Artificial Life (Lecture Notes in Computer Science, Vol. 3630). Berlin: Springer, pp. 210–220.
5.
BuchliJRighettiLIjspeertAJ (2008) Frequency analysis with coupled nonlinear oscillators. Physica D: Nonlinear Phenomena237: 1705–1718.
6.
BuehlerMKoditschekDKindlmannP (1994) Planning and control of robotic juggling and catching tasks. The International Journal of Robotics Research13: 101–118. http://ijr.sagepub.com/content/13/2/101.abstract.
7.
CafutaPCurkB (2008) Control of nonholonomic robotic load. In 10th IEEE International Workshop on Advanced Motion Control, 2008. AMC ’08, pp. 631–636.
8.
CalinonSGuenterFBillardA (2007) On learning, representing, and generalizing a task in a humanoid robot. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics37: 286–298.
9.
CastelliniCvan der SmagtP (2009). Surface EMG in advanced hand prosthetics. Biological Cybernetics100: 35–47.
10.
FilhoACdPDutraMSRaptopoulosLSC (2005) Modeling of a bipedal robot using mutually coupled rayleigh oscillators. Biological Cybernetics92: 1–7. DOI: 10.1007/s00422-004-0531-110.1007/s00422-004-0531-1.
11.
FukuokaYKimuraHCohenAH (2003) Adaptive dynamic walking of a quadruped robot on irregular terrain based on biological concepts. The International Journal of Robotics Research22: 187–202. http://ijr.sagepub.com/content/22/3-4/187.abstract.
12.
FurutaK (2003) Control of pendulum: from super mechano-system to human adaptive mechatronics. In Proceedings 42nd IEEE Conference on Decision and Control, 2003, Vol. 2, pp. 1498–1507.
13.
GamsAIjspeertAJSchaalSLenarcicJ (2009a) On-line learning and modulation of periodic movements with nonlinear dynamical systems. Autonomous Robots27: 3–23.
14.
GamsAPetričTŽlajpahL (2009b) Controlling yo-yo and gyroscopic device with nonlinear dynamic systems. In 18th International Workshop on Robotics, Alpe-Adria-Danube Region, Brasov, Romania, 25–27 May 2009.
15.
GamsAŽlajpahLLenarčičJ (2007) Imitating human acceleration of a gyroscopic device. Robotica25: 501–509.
16.
GangadharGJosephDChakravarthyV (2007) An oscillatory neuromotor model of handwriting generation. International Journal on Document Analysis and Recognition10: 69–84. DOI: 10.1007/s10032-007-0046-010.1007/s10032-007-0046-0.
17.
HakenHKelsoJASBunzH (1985) A theoretical model of phase transitions in human hand movements. Biological Cybernetics51: 347–356. DOI: 10.1007/BF0033692210.1007/BF00336922.
18.
HashimotoK.NoritsuguT. (1996). Modeling and control of robotic yo-yo with visual feedback. In Proceedings 1996 IEEE International Conference on Robotics and Automation, Vol. 3, pp. 2650–2655.
IlgWAlbiezJJedeleHBernsKDillmannR (1999) Adaptive periodic movement control for the four legged walking machine BISAM. In Proceedings 1999 IEEE International Conference on Robotics and Automation, Vol. 3, pp. 2354–2359.
23.
InoueKMaSJinC (2004) Neural oscillator network-based controller for meandering locomotion of snake-like robots. In Proceedings 2004 IEEE International Conference on Robotics and Automation. ICRA ’04, Vol. 5, pp. 5064–5069.
24.
JalicsLHemamiHZhengY (1997) Pattern generation using coupled oscillators for robotic and biorobotic adaptive periodic movement. In Proceedings 1997 IEEE International Conference on Robotics and Automation, Vol. 1, pp. 179–184.
25.
JinH-LYeQZacksenhouseM (2009) Return maps, parameterization, and cycle-wise planning of yo-yo playing. Transactions on Robotics25: 438–445.
26.
JinH-LZackenhouseM (2002) Yoyo dynamics: Sequence of collisions captured by a restitution effect. Journal of Dynamic Systems, Measurement, and Control124: 390–397. http://link.aip.org/link/?JDS/124/390/1 .
27.
JinH-LZacksenhouseM (2003) Oscillatory neural networks for robotic yo-yo control. IEEE Transactions on Neural Networks14: 317–325.
28.
JindaiMWatanabeT (2007) Development of a handshake robot system based on a handshake approaching motion model. In 2007 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 1–6.
29.
KasugaTHashimotoM (2005) Human–robot handshaking using neural oscillators. In Proceedings of the 2005 IEEE International Conference on Robotics and Automation. ICRA 2005, pp. 3802–3807.
30.
KimuraHAkiyamaSSakuramaK (1999) Realization of dynamic walking and running of the quadruped using neural oscillator. Autonomous Robots7: 247–258. DOI: 10.1023/A:100892452154210.1023/A:1008924521542.
31.
KralemannBCimponeriuLRosenblumMPikovskyAMrowkaR (2008) Phase dynamics of coupled oscillators reconstructed from data. Physical Review E77: 066205.
32.
KunAMillerIWT (1996) Adaptive dynamic balance of a biped robot using neural networks. In Proceedings 1996 IEEE International Conference on Robotics and Automation, Vol. 1, pp. 240–245.
33.
LiuCChenQZhangJ (2009) Coupled van der Pol oscillators utilised as central pattern generators for quadruped locomotion. In Proceedings of the Chinese Control and Decision Conference, 2009. CCDC ’09, pp. 3677–3682.
34.
MatsuokaK (1985) Sustained oscillations generated by mutually inhibiting neurons with adaptation. Biological Cybernetics52: 367–376. DOI: 10.1007/BF0044959310.1007/BF00449593.
35.
MatsuokaKOhyamaNWatanabeAOoshimaM (2005) Control of a giant swing robot using a neural oscillator. In Proceedings of ICNC, Vol. 2, pp. 274–282.
36.
MatsuokaKOoshimaM (2007) A dish-spinning robot using a neural oscillator. International Congress Series1301: 218–221.
MorimotoJEndoGNakanishiJChengG (2008) A biologically inspired biped locomotion strategy for humanoid robots: Modulation of sinusoidal patterns by a coupled oscillator model. IEEE Transactions on Robotics24: 185–191.
42.
PetricTGamsAZlajpahL (2009) Modeling and control strategy for robotic Powerball. In 18th International Workshop on Robotics, Alpe-Adria-Danube Region, Brasov, Romania, 25–27 May 2009.
43.
PetričTCurkBCafutaPŽlajpahL (2010) Modeling of the robotic powerball: a nonholonomic, underactuated, and variable structure-type system. Mathematical and Computer Modelling of Dynamical Systems DOI: 10.1080/13873954.2010.48423710.1080/13873954.2010.484237.
44.
ikovskyARosenblumMKurthsJHilbornRC (2002) Synchronization: A universal concept in nonlinear science. American Journal of Physics70: 655–655. http://link.aip.org/link/?AJP/70/655/1.
45.
RighettiLBuchliJIjspeertAJ (2006) Dynamic Hebbian learning in adaptive frequency oscillators. Physica D216: 269–281.
46.
RighettiLIjspeertA (2006) Programmable central pattern generators: an application to biped locomotion control. In Proceedings 2006 IEEE International Conference on Robotics and Automation. ICRA 2006, pp. 1585–1590.
47.
RonsseRLefevrePSepulchreR (2007) Rhythmic feedback control of a blind planar juggler. IEEE Transactions on Robotics23: 790–802.
48.
RonsseRVitielloNLenziTvan den KieboomJCarrozzaMIjspeertA (2010) Human–robot synchrony: flexible assistance using adaptive oscillators. IEEE Transactions on Biomedical Engineering99: 1.
49.
SatoTHashimotoMTsukaharaM (2007) Synchronization based control using online design of dynamics and its application to human–robot interaction. In IEEE International Conference on Robotics and Biomimetics, 2007. ROBIO 2007, pp. 652–657.
50.
SchaalSAtkesonC (1993) Open loop stable control strategies for robot juggling. In Proceedings 1993 IEEE International Conference on Robotics and Automation, pp. 913–918.
51.
SchaalSMohajerianPIjspeertA (2007) Dynamics systems vs. optimal control –a unifying view. Progress in Brain Research165: 425–445. DOI: 10.1016/S0079-6123(06)65027-910.1016/S0079-6123(06)65027-9.
52.
SciaviccoLSicilianoBSciaviccoB (2000) Modelling and Control of Robot Manipulators, 2nd edn.New York: Springer-Verlag Inc.
53.
SentisLParkJKhatibO (2010) Compliant control of multicontact and center-of-mass behaviors in humanoid robots. IEEE Transactions on Robotics26: 483–501.
54.
SpongM (1995) The swing up control problem for the acrobot. IEEE Control Systems Magazine15: 49–55.
55.
SpongMBulloF (2005) Controlled symmetries and passive walking. IEEE Transactions on Automatic Control50: 1025–1031.
56.
StrogatzSH (1986) The Mathematical Structure of the Human Sleep–Wake Cycle. New Yok: Springer-Verlag Inc.
57.
StrogatzSH (1994) Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity). Perseus Books Group.
58.
ThompsonSEPatelRV (1987) Formulation of joint trajectories for industrial robots using b-splines. IEEE Transactions on Industrial Electronics34: 192–199.
59.
TsudaSZaunerK-PGunjiY-P (2007) Robot control with biological cells. Biosystems87: 215–223.
60.
UdeAAtkesonCGRileyM (2000) Planning of joint trajectories for humanoid robots using b-spline wavelets. In Proceedings of ICRA, pp. 2223–2228.
61.
van der PolB (1934) The nonlinear theory of electric oscillations. Proceedings of the Institute of Radio Engineers22: 1051–1086.
62.
VeskosPDemirisY (2005) Developmental acquisition of entrainment skills in robot swinging using van der Pol oscillators. Lund University Cognitive Studies123: 87–93. http://cogprints.org/4969/.
63.
ŽlajpahL (2006) Robotic yo-yo: modelling and control strategies. Robotica24: 211–220.
64.
WilliamsonM (1999) Designing rhythmic motions using neural oscillators. In Proceedings. 1999 IEEE/RSJ International Conference on Intelligent Robots and Systems. IROS ’99, Vol. 1, pp. 494–500.
65.
WilliamsonMM (1998) Neural control of rhythmic arm movements. Neural Networks11: 1379–1394.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
