Mechatronics 22 (2012) 76–82

Contents lists available at SciVerse ScienceDirect

Mechatronics

journal homepage: www.elsevier .com/ locate /mechatronics

Multipendulum mechatronic setup: Design and experiments q

Alexander L. Fradkov a,b, Boris Andrievsky a,b,⇑, Kirill B. Boykov c

a Institute for Problems of Mechanical Engineering of the Russian Academy of Sciences, 61, V.O. Bolshoy Av., Saint Petersburg 199178, Russiab National Research University of Information Technologies, Mechanics and Optics, Saint Petersburg, Russiac Corporation Granit-7, Saint Petersburg, Russia

a r t i c l e i n f o

Article history:Received 22 March 2011Accepted 20 November 2011Available online 16 December 2011

Keywords:Nonlinear dynamicsCommunication constraintsMechatronic setup

0957-4158/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.mechatronics.2011.11.006

q Preliminary version was presented at the Sixth EURConference (ENOC 2008), June 30–July 4, 2008, Saint⇑ Corresponding author at: Institute for Problems

the Russian Academy of Sciences, 61, V.O. Bolshoy ARussia.

E-mail addresses: alexander.fradkov@gmail.coandrievsky@gmail.com (B. Andrievsky), kirill.boykow

a b s t r a c t

In the paper a novel multipendulum mechatronic setup is described. The setup may be used for imple-menting different algorithms of estimation, synchronization and control and solving various researchand educational tasks in hybrid modeling, analysis, identification and control of mechanical systems. Italso allows one to study influence of the data communication protocols on dynamics of distributed mech-atronic complexes. Experimental results confirming existence of the lower bound for transmission rateensuring reasonable estimation accuracy under data rate constraints are presented.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Control and synchronization of oscillatory mechanical systems isof theoretical interest and has many practical applications. Theproblems of control and synchronization of nonlinear oscillatingprocesses demand development of new sophisticated algorithms,which should be experimentally tested. For the purposes of researchand control engineering education it is important to build up appro-priate laboratory equipment and software for investigation of thiskind of system. There are many papers where this problem was con-sidered and significant results have been achieved [9,30,15,10,11].During last decades various mechatronic laboratory setups weredescribed in the literature: inverted pendulums [17,8], reaction-wheel pendulum [31], cart-pendulum [18,19], Furuta pendulum[33,20], coupled two-pendulum systems [6,22,15,35], metronomeson a moving base [28,27], pendulum-like juggling system [32], etc.A promising design described in [29] provides a possibility to modela variety of different oscillatory systems in a single setup.

However there is still a demand for equipment allowing to testand study new control and data exchange algorithms fro complexdistributed systems. In the present paper the novel multipendulummechatronic setup – the multipendulum mechatronic setup of theInstitute for Problems of Mechanical Engineering (MMS IPME) isdescribed.

ll rights reserved.

OMECH Nonlinear DynamicsPetersburg, Russia.of Mechanical Engineering ofv., Saint Petersburg 199178,

m (A.L. Fradkov), boris.@gmail.com (K.B. Boykov).

MMS IPME includes a modular multi-section mechanical oscil-lating system, electrical equipment (with the computer interfacefacilities) and a personal computer for experimental data process-ing, representation of the results and real-time control. Themechanical system consists of the pendulums connected by meansof the springs. A key feature of MMS IPME is the possibility tochange the number of DOF in a broad range. Starting from a singlependulum one may increase the number of the pendulums in thechain up to 50. It is also possible to connect pendulums into a mul-ti-chain (network) configuration.

MMS IPME allows one to implement different algorithms ofestimation, synchronization and control. Particularly, it may beused as a multi-degree of freedom mechanical model of controlledphysical systems [2] and power system networks [21]. This setupmay also find extensive educational applications as an exampleof a complex dynamical system equipped by local and global con-trollers and computer data exchange facilities. It can be used toexperimentally test and validate theoretical results.

In Section 2, a brief description of the mechanical part is pre-sented. For making laboratory experiments and on-line control,electrical design, data exchange interface and software tools werecreated. Their description is given in Section 3. Possible ways forimproving the data exchange in multiagent mechatronic com-plexes are outlined in Section 4, where both hardware-softwareand algorithmic solutions are presented. Some results of experi-ments with MMS IPME are described in Section 5.

2. Design of mechanical part

The schematic of a single pendulum section is presented inFig. 1 (see also photo in Fig. 2). The foundation of the section is a

Fig. 1. Schematics of the pendulum section. Effective length can be changed byreplacing counterweights.

Fig. 2. Photo of the chain of 12 pendulum sections and the motor.

A.L. Fradkov et al. / Mechatronics 22 (2012) 76–82 77

hollow rectangular body. An electrical magnet and electronic con-troller board are mounted inside the body. The support containingthe platform for placing the sensors in its middle part is mountedon the foundation. The pendulum itself has a permanent magnettip in the bottom part. The working ends of the permanent magnetand the electrical magnet are posed exactly opposite each otherand separated with a non-magnetic plate in the window of thebody. The idea behind control of the pendulum is changing thepoles of the electrical magnet by means of switching the directionof the current in the windings of the electrical magnet. Addition-ally, two computer-controlled electrical motors may be connectedwith the first and the last pendulums of the chain via the torsionsprings for changing the boundary conditions of the chain.1 In or-der to change the eigenfrequency of the pendulum oscillations thependulum is endowed with additional counterweight changing itseffective length (the distance between the suspension point andthe center of mass). On the rotation axis of the pendulum the opticalencoder disk for measuring the angle (phase) of the pendulum ismounted. It has 90 slits. The peripheral part of the disk is posed intothe slit of the sensor support. The sensor consists of a radiator (emit-ting diode) and a receiver (photodiode). The obtained sequences of

1 At present, only the ‘‘left-side’’ motor is in service. The boundary conditions of the‘‘right’’ end of the chain are free.

signals allow one to measure the pendulum angle, evaluate ampli-tude of oscillations and register time instants of crossing the lowerequilibrium position.

The axes of the neighboring sections are connected with the tor-sion springs, arranging force interaction and allowing exchangingenergy between the pendulums. The set of interconnected pendu-lum sections represents a complex oscillatory dynamical system,characterized by nonlinearity and high number of degrees of free-dom. In principle, any number of sections can be connected. At themoment mechanical parts of 50 sections are manufactured.

3. Hardware and software of MMS IPME

The control system of the complex consists of: controllers forpendulum modules and motors; the supervisory computer andthe interface unit; the electric power devices (power amplifiers,power supply units); the communication channel.

3.1. Local controllers of modules

Controllers of the pendulum modules and the motors aremounted at the universal logic board (ULB) developed for unifica-tion of modules schematic and achievement of the highest possiblecontroller speed. The kernel of the ULB is the erasable programma-ble logic device EPM240T100C5 manufactured by Altera Corpora-tion. Each controller is a specialized firmware machine withbuilt-in microprogram for control of pendulums or DC motors,and for measuring the shaft rotation angles. The ULB also includesthree input signal shapers (comparators), mounted on the mechan-ical parts of the modules, the secondary supply sources and thetimer. The firmware is designed with the help of the design systemQuartus II [1].

The local controllers are used for offloading the upper levelsupervisory computer, formation of the pulse-width modulatedcontrol signal, the shaft angle measurement and implementationof the low-level data exchange functions.

3.2. Communication protocol

The communication protocol ensures transferring the instruc-tions and command qualifiers to the interface board of the pendu-lum sections and collects the data from the interface board sensors.The communication protocol is based on explicit addressing withusage of the read-write registers of the Enhanced Parallel Port. Log-ical separation between the data and instruction flows is madewith usage of two highest stages of the address register. The restfive stages of this register contain the module address for theread-write cycle.

78 A.L. Fradkov et al. / Mechatronics 22 (2012) 76–82

The described data exchange protocol ensures exchange rate upto 9615 Hz for simultaneous operation over the bus with the totalnumber (52) of modules, and up to 500 kHz for operation with asingle module.

3.3. The supervisory computer and the interface unit

The personal computer with Intel Celeron processor and operat-ing system GNU/Linux is chosen for upper-level control of the localcontrollers, processing and visualization of experimental data.Operating system GNU/Linux allows to optimize the CPU time allo-cation in favor of the priority tasks of query and control.

For supporting the data exchange operations, the packagescomedi 0.7.76 and comedilib 0.8.1 are installed additionally to OSALT Linux Desktop 4.0 Personal. These packages represent a special-ized set of software for abstraction from the hardware peculiaritiesby means of a virtual file system. The package comedi 0.7.76 is a setof drivers supporting the universal analog-to-digital and digital in-put/output devices of the leading manufacturers such as Advan-tech, National Instruments, ComputerBoards.

The experimental results may be postprocessed using the pack-ages MATLAB (running under GNU/ Linux) or its open-source analogSCILAB. Alternatively, one can process experimental data using pack-age Octave or package gnuplot for visualization.

4. Directions for optimization of communication channel formultiagent mechatronic complexes

The first experiments with MMS IPME consisting of fivependulum modules and one motor have showen correctness ofthe technical solutions used for control of a multiagent distributedmechatronic system. However some problems have been alsorevealed, caused by a single-computer implementation of thehardware–software system where the single PC is used both forequipment control and for the real-time visualization of theexperiments.

4.1. Hardware–software solutions

The conventional way for improving the system performance isusing the real-time support modules of higher execution prioritysuch as rtai or rtLinux. This solution would help stabilize the dataacquisition process. However, in the case of programmable input/output, the problem of increasing the sampling rate remains achallenging one.

A promising solution for modernization of the mechatronic set-up would be changeover from the existing single-computer archi-tecture to the multi-computer complex. In this case, the differentnodes of the complex may be linked over the network Ethernetbased on a synchronous access protocol, without any associationof the complex’ network with the existing computer networks toavoid interfering on the order of messaging for protocols ARP orICMP. Hardware solution of the controller for data exchange withthe stand modules (assuming allocation one controller for 3–5modules) is reasonable to realize on the base of processor modulesx86 for ensuring the uniform cycle of support and developing thesoftware of the complex.

Operating systems microLinux (ucLinux), Embedded Linux, orOpenWrt with built-in support for the real-time mode are plannedto use for exchange controllers. An advantage of modules CPC150and CPC109 is a compact solution (single-board computer withintegrated digital I/O). There is an additional opportunity in connec-tion the standard monitor and keyboard to the modules CPC150 andCPC109 to perform stand-alone operation without the upperlevel computer. Transition from a centralized architecture to the

distributed one significantly increases the computational efficiencyof the complex, allowing to flexibly resource allocation betweencomputing modules.

4.2. Algorithmic solutions. State estimation under the data ratelimitations

The abovementioned directions for increasing the performancecapability and the data exchange rate of the complex may be ref-fered to as hardware–software solutions. During the last decadethe considerable attention was riveted to the algorithmic solutionsfor improving the system performance under the data-rate limita-tions, see the surveys [26,5], the monograph [24] and the refer-ences therein. Particularly, it has been shown that the control/observation of linear systems is possible if and only if the capacityof the information channel exceeds the entropy production of thesystem at the equilibrium (the Data Rate Theorem) [25]. The cod-ing-decoding schemes were proposed giving an opportunity toget closer to the minimum possible data rate. Two ideas are basi-cally applied for this purpose: using smart sensors incorporatingthe model of the observed plant dynamics, and also applying thezooming strategy, where the range of the encoder is updated dur-ing the control or observation process [7,23,34].

First results on synchronization of nonlinear systems underinformation constraints are presented in [12], where the so calledobserver-based synchronization scheme is employed. It is shownthat for the first-order coder-decoder scheme the upper bound oflimit synchronization error is proportional to the maximum rateof the coupling signal and inversely proportional to the informa-tion transmission rate (channel capacity). The controlled synchro-nization problem was analyzed in [3]. It was shown that in the caseof an ideal channel and non-corrupted measurements, the outputfeedback controlled synchronization strategy with the full orderencoder/decoder pair, ensures exponentially vanishing synchroni-zation error if the channel capacity exceeds a certain threshold.This concurs with the known results, obtained for linear systemsin [34,7].

Further on, the approach of Fradkov et al. [12] is applied to theobservation of nonlinear systems over the limited-band communi-cation channel in [13]. Some experimental results on application ofthis approach to feedback control over the limited capacity com-munication channel are presented in [14]. Let us briefly recall theabovementioned observation schemes.

Consider the following nonlinear plant model:

_xðtÞ ¼ AxðtÞ þ BwðyÞ; yðtÞ ¼ CxðtÞ; ð1Þ

where xðtÞ 2 Rn is the vector of plant state variables; y(t) is the sca-lar output variable; A is an (n � n)-matrix; B is an (n � 1)-matrix; Cis an (1 � n)-matrix, w(y) is a continuous nonlinearity.

The problem is to obtain the state estimation of (1) over the dig-ital communication channel with the limited bandwidth. The mea-sured data are sampled with a certain sampling rate Ts and arerepresented by finite-length codewords to be transmitted overthe channel. The following uniform memoryless (static) quantizeris employed:

qm;MðyÞ ¼d � hd�1yi; if jyj 6 M;

M signðyÞ; otherwise;

(ð2Þ

where M > 0 is a real number (the quantizer range), m 2 Z is a posi-tive integer, d = 21�mM; h�i denotes the round-up to the nearest inte-ger, sign(�) is the signum function. The quantization interval [�M,M]is equally split into 2m parts. Therefore, the cardinality of the map-ping qm,M image is equal to 2m + 1 and each codeword containsR = log2(2m + 1) = log2(2M/d + 1) bits. The one-step memory coderuses the central number c[k], k = 0,1, . . . with the initial condition

A.L. Fradkov et al. / Mechatronics 22 (2012) 76–82 79

c[0] = 0 [34]. At the step k, the coder compares the current mea-sured output y[k] with the number c[k], forming the deviation sig-nal @y[k] = y[k] � c[k]. Then @y[k] is discretized with the given m andM = M[k] according to (2). The quantized output signal

�@y½k� ¼ qm;M½k�ð@y½k�Þ ð3Þ

is represented as an R-bit codeword and transmitted over the com-munication channel to the decoder. At the next step, the centralnumber c[k + 1] and the quantizer range M[k] are renewed by thefollowing update algorithms [13]:

c½kþ 1� ¼ c½k� þ �@y½k�; c½0� ¼ 0; k ¼ 0;1; . . . ; ð4ÞM½k� ¼ ðM0 �M1Þqk þM1; k ¼ 0;1; . . . ; ð5Þ

where 0 < q 6 1 is the decay parameter, M1 stands for the limit va-lue of M[k]. The initial value M0 should be large enough to captureall the region of possible values of y[0].

The coder of the full order embeds the observer. In [14], theobservation error (innovation signal) is transmitted over the channelrather than a measured plant output. For describing a such kind ofthe coders, let us introduce the error between the plant and obser-ver outputs as eðtÞ ¼ yðtÞ � yðtÞ ¼ CeðtÞ. This signal is subjected tothe coding procedure (2)–(5) instead of y(t), forming the quantizedsignal �e½k�. The following state estimation algorithm is imple-mented at the coder:

_xðtÞ ¼ AxðtÞ þ BwðyÞ þ L�eðtÞ; yðtÞ ¼ CxðtÞ;�eðtÞ ¼ �e½k� as t 2 ½tk; tkþ1Þ; tk ¼ kTs; k ¼ 0;1;2; . . . :

ð6Þ

where x 2 Rn stands for the estimate of the plant state vector x(t),(n � 1)-matrix L is observer gain (design parameter).

5. Experiments with MMS IPME

The system is currently in a limited operation mode. Five activeand up to 46 passive sections are ready for connection. Already atthis stage the system can be used for demonstration and for re-search. In one of the series of experiments, either in phase or anti-phase synchronization of the chain of pendulums, excited by theexternal harmonic torque, is demonstrated. Another series ofexperiments was carried out for evaluation of the data transmis-sion schemes of Section 4.2. Some results are presented below.

5.1. Modeling the chain of pendulums

Following [4], the rotation angle of the drive shaft connectedwith the first pendulum of the chain is considered as the systeminput. The last (Nth) pendulum in the chain is mechanically con-nected only with the previous one, no boundary conditions forNth pendulum are specified. This leads to the following model ofthe chain dynamics:

2 3 4 5 60

0.2

0.4

0.6

0.8A1(ω), A2(ω), A3(ω), A4(ω)

ω, r

Fig. 3. Frequency magnitude responses for pendulum rotation angles in the chain (aA4(xm)–dash-dot line.

€u1 þ q _u1 þx20 sinu1 � kðu2 � 2u1Þ ¼ kuðtÞ;

€ui þ q _ui þx20 sinui � kðuiþ1 � 2ui þui�1Þ ¼ 0;ði ¼ 2;3; . . . ;N � 1Þ;

€uN þ q _uN þx20 sin uN � kðuN �uN�1Þ ¼ 0;

8>>><>>>:

ð7Þ

where ui = ui (t) (i = 1, 2, . . . N) are the pendulum deflection an-gles; u = u(t) is the controlling action (the rotation angle of the driveshaft). The values q,x0,k are the system parameters: q is the viscousfriction parameter; x0 is the natural frequency of small oscillationsof the isolated pendulum; k is the coupling strength parameter,which depends on the stiffness of the connecting springs. The model(7) parameters were estimated based on the mass-geometry prop-erties of the mechanical system. Then the obtained values were cor-rected by means of the trial-and-error procedure applied to theexperimental data sets. The following parameter estimates were fi-nally obtained: x0 = 5.5 s�1,q = 0.95 s�1, k = 5.8 s�2. Note that if thependulums are connected in 2D grid then (7) should be modifiedaccording to the coupling topology described by Laplacian matrixof the network graph.

5.2. Synchronization in the harmonically excited chain of pendulums

The series of experiments has been fulfilled for validation of theanalytical results given in [4] related to synchronization of the pen-dulums chain, excited by means of the external harmonic torque.

The chain of four pendulums was taken. Following [4], model(7) was linearized in the neighborhood of the equilibrium. The fre-quency magnitude responses Ai(xm) (i = 1, . . . ,4) from the driveshaft rotation angle u(t) to angular deflections ui(t) are evaluatednumerically and plotted in Fig. 3. The phase shifts Dwi,i+1(xm) be-tween the adjacent pendulums are depicted in Fig. (4). It is seenfrom the plots that the pendulums demonstrate inphase synchro-nization if the excitation frequency xm is small enough. Approxi-mately antiphase synchronization appears for sufficiently largexm. It is also seen that the pendulum chain may play a role of amechanical bandpass filter for intermediate frequencies.

The corresponding experimental results are demonstrated inFigs. 5–7, where the time histories of the pendulums’ rotationangles in the steady-state mode are plotted. An inphase synchroni-zation in the chain of pendulums for the case xm = 4.0 rad/s is pre-sented in Fig. 5. An antiphase synchronization for xm = 8.9 rad/s isseen in Fig. 6. The running wave for the case xm = 6 rad/s is demon-strated in Fig. 7. The similar results are obtained by experimentswith the chain of twelve pendulum sections.

5.3. Testing the data exchange and the state estimation algorithm

The state estimation scheme of Section 4.2 (Eqs. (2)–(6)) ismodified for taking into account presence of the external excitationsignal u(t), applied to the chain. In our experiments, the signal u(t)

7 8 9 10ad/s

nalytical evaluation). A1(xm)–solid line, A2(xm)–dashed line, A3(xm)–dotted line,

2 3 4 5 6 7 8 9 10−200

−150

−100

−50

0Δψm,1(ω), Δψ1,2(ω), Δψ2,3(ω), Δψ3,4(ω), deg

ω, rad/s

Fig. 4. Phase shifts between the pendulum rotation angles in the chain (analytical evaluation). Dwm,1(xm)–solid line, Dw1,2(xm)–dashed line, Dw2,3(xm)–dotted line,Dw3,4(xm)–dash-dot line.

30 30.5 31 31.5 32 32.5 33 33.5 34 34.5 35−200

−100

0

100

200φm(t), φ1(t), φ2(t), deg

t, s

Fig. 5. Experimental result. In phase synchronization in the chain of pendulums. um(t)–solid line, u1(t)–dashed line, u2(t)–dotted line; xm = 4.0 rad/s.

30 30.5 31 31.5 32 32.5 33 33.5 34 34.5 35−500−400−300−200−100

0100200300400500

φm(t), φ1(t), φ2(t), deg

t,s

Fig. 6. Experimental result. Antiphase synchronization in the chain of pendulums. um(t)–solid line, u1(t)–dashed line, u2(t)–dotted line; xm = 8.9 rad/s.

30 30.5 31 31.5 32 32.5 33 33.5 34 34.5 35−200

−100

0

100

200φm(t), φ1(t), φ2(t), φ3(t), φ4(t), deg

t, s

Fig. 7. Experimental result. Running wave in the chain of four pendulums. um(t)–solid line, u1(t)–dashed line, u2(t)–dotted line, u3(t)–dash-dot line, u4(t)–thin line;xm = 6.0 rad/s (xm lies inside the pass band).

80 A.L. Fradkov et al. / Mechatronics 22 (2012) 76–82

0 100 200 300 400 500 6000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

R, bit/s

Q(R,ν)

ν=8

R8R6R4

ν=6ν=4

Fig. 8. Relative estimation error vs transmission rate R for different m. Solidline–m = 4, dashed line–m = 6, dash-dot line – m = 8. Rm – lover bound for m = 4,6, 8.

0 1 2 3 4 5

−1

0

1

φ1,exp, φ1,est, ε1, rad

t, s

0 5 10 15 20 25 30−10

−5

0

5

10

15

φ1,exp, φ1,est, ε1, rad

t, s

Fig. 9. Measured process u1(t) (dashed line), the estimate u1ðtÞ, obtained by thedecoder (solid line), the estimation error e1ðtÞ ¼ u1ðtÞ � u1ðtÞ (dotted line).Ts = 0.020 s, m = 4,R = 200 bit/s. a)– case of the harmonic excitation, b) – case ofthe irregular excitation.

0 1 2 3 4 5

−10

0

10

dφ1,est/dt, rad/s

t, s

0 5 10 15 20 25 30−20

−10

0

10

20dφ1,est/dt, rad/s

t, s

Fig. 10. Estimate of the angular rate _u1ðtÞ, obtained by the decoder. Ts = 0.020 s,m = 4,R = 200 bit/s. a) – case of the harmonic excitation, b) – case of the irregularexcitation.

A.L. Fradkov et al. / Mechatronics 22 (2012) 76–82 81

are measured by the optical encoder with the accuracy of 2 degreesand 100 Hz sampling frequency. The measured data are transmit-ted over the communication link without any restrictions on thedata rate. Therefore, for the considered system, the autonomousplant model (1) is replaced by the following exogenous model:

_xðtÞ ¼ AxðtÞ þ BwðyÞ þ DuðtÞ; yðtÞ ¼ CxðtÞ; ð8Þ

where uðtÞ 2 Rm stands for the external input, D is an (n �m) ma-trix, the other notations are the same as in (1). Respectively, equa-tion of the modified observer (6) reads as

_xðtÞ ¼ AxðtÞ þ BwðyÞ þ D~uðtÞ þ L�eðtÞ; yðtÞ ¼ CxðtÞ;�eðtÞ ¼ �e½k� as t 2 ½tk; tkþ1Þ; tk ¼ kTs; k ¼ 0;1;2; . . . :

ð9Þ

where ~uðtÞ denotes the measured value of the exogenous input sig-nal u(t). The errors of the plant input u(t) and output y(t) measure-ments introduce imperfections into the data transmissionprocedure and impose limitation on the estimation accuracy, whichis achievable in the real-world systems.

The state estimation procedure Eq. (2), (3), (4), (5), (9) is testedas applied to the chain of four pendulum sections, excited by themotor, which is connected with the pendulum #1 via the torsionspring. Harmonic and irregular input voltages is applied to the mo-tor in the different experiments. The first-order data transmissionscheme (2)–(5) is used for transferring the rotation angles of themotor and the pendulum #4 to the central computer. For obtainingthe estimates of the angles and angular velocities for the pendu-lums ##1–3, the full-order coding algorithm (2), (4), (5), (6) is ap-plied. The coder/decoder parameters are taken as follows:Ts 2 [0.01,0.05] s, m 2 {4,6,8}, M0,m = 15, M1m = 0.1,M0,e = 2,M1,e = 5�10�4, q = 0.99,L = [27.3,362]T. The estimation accuracyhas been calculated as the mean-square relative error QðR; mÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR T

0ðu1�u1Þ2 dtR T

0u2

1 dt

s; T ¼ 30 s. The experimental results are depicted in

Figs. 8–10. It is seen that the data transmission rate may be takenabout 200 bit/s, ensuring the reasonable accuracy of the statetransmission in approximately 3% in the terms of the relativemean-square error. It is also demonstrated that there exists a lowerbound Rm of the data transmission rate, such that remote state esti-mation is not possible for smaller rates. This confirms the results ofFradkov et al. [14].

6. Conclusions

A novel multipendulum mechatronic setup is presented. Thesetup is aimed at solving various research and educational tasksin the areas of hybrid systems modeling, analysis, identificationand control of mechanical systems as well the data communicationin distributed mechatronic complexes.

Experiments with the complex of twelve interconnected pendu-lums and the electric motor, exciting the system, have shown thatthe type of the synchronization mode (inphase or antiphase)depends on the excitation frequency. This result agrees with the

82 A.L. Fradkov et al. / Mechatronics 22 (2012) 76–82

conclusions of Andrievsky and Fradkov [4]. Possible ways forimproving the data exchange in multiagent mechatronic com-plexes are outlined. The data transmission scheme of Fradkovet al. [14] has been modified for non-autonomous systems andexperimentally tested. It is seen, that the data transmision ratemay be taken about 200 bit/s, ensuring the appropriate accuracyof the state transmission over the digital communication channel.It is also demonstrated that there exists a lower bound of theadmissible data transmission rate which confirms the results ofFradkov et al. [14].

The future work is aimed at studying the different kinds of syn-chronization and nonlinear waves propagation in multi-degree offreedom dynamical systems. Another avenue of research is imple-mentation of different global and local control schemes taking intoaccount bandwith limitations, signal drops and delays in the com-munication channel.

Acknowledgments

Partially supported by the Dutch–Russian program on interdis-ciplinary mathematics ‘‘Dynamics and Control of Hybrid Mechani-cal Systems’’ (Project NWO–RFBR 047.011.2004.004), the RussianFoundation for Basic Research (Project No. 09-08-00803, 11-08-01218), the Russian Federal Program ‘‘Research and TeachingCadres’’, contracts NN 16.740.11.0042, 14.740.11.0942, and bythe Program of Basic Research of OEMMPU RAS No 2 ‘‘Controland Safety in Energy and Technical Systems’’.

The mechanical part of the setup is designed by B.P. Lavrov [16].The authors are grateful to Prof. Henk Nijmeijer and Prof. Alex-

ander Pogromsky, both from Eindhoven University of Technology,for their seminal cooperation in the Dutch–Russian NWO–RFBR re-search program.

References

[1] Quartus II Software Version 9.1 SP2 Release Notes. Altera Corporation, April2010. <http://www.altera.com/literature/rn/rn_qts.pdf.

[2] Aero EL, Fradkov AL, Andrievsky B, Vakulenko SA. Dynamics and control ofoscillations in a complex crystalline lattice. Phys Lett A 2006;353(1):24–9.

[3] Fradkov AL, Andrievsky B, Evans RJ. Synchronization of passifiable Luriesystems via limited-capacity communication channel. IEEE Trans Circuits Syst I2009;56(2):430–9.

[4] Andrievsky B, Fradkov AL. Behavior analysis of harmonically forced chain ofpendulums. In: IEEE international conference on control applications (CCA’09),Saint Petersburg, Russia, July 8–10, 2009. p. 1563–7.

[5] Andrievsky BR, Matveev AS, Fradkov AL. Control and estimation underinformation constraints: toward a unified theory of control, computation andcommunications. Autom Remote Control 2010;71(4):572–633.

[6] Bennett M, Schatz M, Rockwood H, Wiesenfeld K. Huygens’s clocks. Proc RoySoc, A: Math Phys Eng Sci 2002;458:563–79.

[7] Brockett RW, Liberzon D. Quantized feedback stabilization of linear systems.IEEE Trans Autom Control 2000;45:1279–89.

[8] Chiu Chih-Hui, Lin You-Wei, Lin Chun-Hsien. Real-time control of a wheeledinverted pendulum based on an intelligent model free controller. Mechatronics2011;21(3):523–33.

[9] Christini DJ, Collins JJ, Linsay PS. Experimental control of high dimensionalchaos: the driven double pendulum. Phys Rev E 1996;54(5).

[10] Fradkov AL. Application of cybernetical methods in physics. Phys – Uspekhi2005;48(2):103–27.

[11] Fradkov AL. Cybernetical physics: from control of chaos to quantumcontrol. Berlin–Heiderlberg: Springer-Verlag; 2007.

[12] Fradkov AL, Andrievsky B, Evans RJ. Chaotic observer-based synchronizationunder information constraints. Phys Rev E 2006;73:066209.

[13] Fradkov AL, Andrievsky B, Evans RJ. Hybrid quantised observer for multi-input-multi-output nonlinear systems. In: Proc 17th IEEE internationalconference on control applications (CCA 2008). p. 1195–1200, San Antonio,Texas, USA, September 3–5, 2008.

[14] Fradkov AL, Andrievsky B, Peaucelle D. Estimation and control underinformation constraints for LAAS helicopter benchmark. IEEE Trans ControlSyst Technol 2010;18(5):1180–7.

[15] Fradkov AL, Andrievsky BR, Boykov KB. Control of the coupled doublependulums system. Mechatronics 2005;15:1289–303.

[16] Fradkov AL, Andrievskiy B, Boykov KB, Lavrov BP. Multipendulum mechatronicsetup for studying control and synchronization. In: Leonov G, NijmeijerH,Pogromsky A, Fradkov A, editors. Dynamics and control of hybridmechanical systems, series on nonlinear science, series B, vol. 14. Singapore:World Scientific; 2010. p. 211–2.

[17] Furuta K,Yamakita M. Swing up control of inverted pendulum. In: Proceedingsof the 1991 international conference on industrial electronics, control andinstrumentation (IECON ’91), October 28–November 1, 1991. IEEE, Piscataway,N.J., Kobe, Japan; 1991. p. 2193–8.

[18] Gawthrop PJ, McGookin E. A LEGO-based control experiment. IEEE Control SystMagn 2004;24(5):43–56.

[19] Graichen K, Treuer M, Zeitz M. Swing-up of the double pendulum on a cart byfeed forward and feedback control with experimental validation. Automatica2007;43(1):63–71.

[20] La Hera PX, Freidovich LB, Shiriaev AS, Mettin U. New approach for swingingup the Furuta pendulum: theory and experiments. Mechatronics2009;19(8):1240–50.

[21] Hill DJ, Chen G. Power systems as dynamic networks. In: Proceedings of IEEEinternational symposium on circuits and systems (ISCAS 2006), Island of Kos,Greece, May 21–24, 2006. IEEE. p. 722–5.

[22] Kumon M, Washizaki R, Sato J, Kohzawaa R, Mizumoto I, Iwai Z. Controlledsynchronization of two 1-DOF coupled oscillators. In: Proceedings of the 15thtriennial world congress of IFAC. Barcelona, Spain; 2002.

[23] Liberzon D. Hybrid feedback stabilization of systems with quantized signals.Automatica 2003;39:1543–54.

[24] Matveev AS, Savkin AV. Estimation and control over communicationnetworks. Boston: Birkhäuser; 2009 (Series: Control Engineering).

[25] Nair GN, Evans RJ. Exponential stabilisability of finite-dimensional linearsystems with limited data rates. Automatica 2003;39:585–93.

[26] Nair GN, Fagnani F, Zampieri S, Evans RJ. Feedback control under data rateconstraints: an overview. Proc IEEE 2007;95(1):108–37.

[27] Oud WT, Nijmeijer H, Pogromski AY. A study of Huijgens’ synchronization:Experimental results. In: Kristin Y. Pettersen, Jan Tommy Gravdahl, HenkNijmeijer, editors. Group Coordination and Cooperative Control. Springer;2006. p. 133–8.

[28] Pantaleone J. Synchronization of metronomes. Am J Phys 2002;70(10):992–1000.

[29] Rijlaarsdam DJ, Pogromski AY, Nijmeijer H. Synchronization between coupledoscillators: a uniform experimental approach. In: Proceedings of the sixthEUROMECH nonlinear dynamics conference (ENOC 2008), June 30–July 4,Saint Petersburg, Russia; 2008. IPACS. <http://lib.physcon.ru/?item=1705>.

[30] Santoboni G, Pogromsky A Yu, Nijmeijer H. Application of partial observabilityfor analysis and design of synchronized systems. Chaos, Solitons Fract2003;13(1):356–63.

[31] Spong MW, Corke P, Lozano R. Nonlinear control of the reaction wheelpendulum. Automatica 2001;37:1845–51.

[32] Suzuki S, Furuta K, Shiratori S. Adaptive impact shot control by pendulum-likejuggling system. Int J Ser C-Mech Syst Mach Elem Manuf 2003;46(3):973–81.

[33] Suzuki S, Pan Y, Furuta K, Hatakeyama S. VS-control with time-varying slidingsector – design and application to pendulum. Asian J Control 2004;6(3):307–16.

[34] Tatikonda S, Mitter S. Control under communication constraints. IEEE TransAutom Control 2004;49(7):1056–68.

[35] Yagasaki K. Extension of a chaos control method to unstable trajectories oninfinite- or finite-time intervals: experimental verification. Phys Lett A2007;368(3 and 4):222–6.