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Modeling and Control of A Cable-Driven Series Elastic Actuator WulinZou1,2,NingboYu1,2 1.InstituteofRoboticsandAutomaticInformationSystems,NankaiUniversity,Tianjin300353,P.R.China 2.TianjinKeyLaboratoryofIntelligentRobotics,NankaiUniversity,Tianjin300353,P.R.China E-mail:[email protected],[email protected] Abstract: Serieselasticactuators(SEA)areplayinganincreasinglyimportantroleinthefieldsofphysicalhuman-robotinteraction. This paperfocusesonthemodelingandcontrolofacable-drivenSEA.First,theschemeofthecable-drivenSEAhasbeenproposed,and avelocitycontrolledDCmotorhasbeenusedasitspowersource. Basedonthis,themodelofthecable-drivenSEAhasbeenbuilt up. Further,atwodegreesoffreedom(2-DOF)controlapproachhasbeenemployedtocontroltheoutputtorque. Simulationresults 7 haveshownthatthe2-DOFmethodhasachievedbetterrobustperformancethanthePDmethod. 1 0 KeyWords:Serieselasticactuator,Stabilizing2-DOFcontroller,Physicalhuman-robotinteraction,Torquecontrol 2 n a 1 Introduction and noise. Horowitz revealed that the 1-DOF configuration J isunabletocopewiththetwoproblemsofachievingadesired 4 Series elastic actuator was first recommended by Pratt and trackingresponseandattainingagooddisturbance/noiserejec- 1 Williamson for its benefits of greater shock tolerance, accu- tionatthesametime,butthe2-DOFcontrollercouldachieve rate and stable force output and the ability of energy storage ] thesetwogoalssimultaneously[17]. Thereasonisthatthe2- Y [1]. Nowadays, SEAs are widely applied in various physical DOFcontrollerprovidesanindependentwaytodesigntrack- human-robotinteractionapplications, e.g., theSEAforwalk- S ing response and optimize disturbance/noise rejection. Sys- ing robots [2], the compact rotary SEA for human assistive . tematic introduction and design procedures about the 2-DOF s robots [3], the compact compliant actuator for rehabilitation c controlmethodcanbefoundin[18,19]. Thispaperisalsoan [ robots[4],etc. extensiveworkof[20–22]. Cable actuation has attracted intensive research for its ad- 1 Thispaperisorganizedasfollows. Section2describesthe vantagesoflowinertia,flexibleinstallation,remoteactuation, v conceptandmodelofthevelocitysourcedcable-drivenSEA. 3 etc. Cables with low weight to length ratio can change the DetailsforthetorquecontrollerdesignaregiveninSection3. 1 force direction intentionally and easily, enable power trans- Simulations and results are presented in Section 4. Section 5 9 mission to a remote distance with less energy loss and space concludesthepaper. 3 occupation [5, 6], and allow detachment of the actuation mo- 0 2 TheCable-DrivenSEA tor from the robot frame [7]. Besides, a cable actuated sys- . 1 temcanbeasafesolutionduetoitsunidirectionalforcecon- 2.1 TheModeloftheCable-DrivenSEA 0 straint and property of breaking when the tension exceeding 7 The principle of a cable-driven SEA is shown in Fig. 1. A the threshold. Many cable actuated systems have been ap- 1 cable-spring series structure is introduced between the motor : pliedforphysicalhuman-robotinteraction,suchastheLOwer v andtheload. Thepowersourceoftheactuatorissuppliedby extremity Powered ExoSkeleton (LOPES) [7], the Universal i avelocitycontrolledDCmotor. X HapticDrive(UHD)forupperextremityrehabilitation[8],the r MR-compatiblewristrobot[9],etc.  a  Theforce/torquecontrollerofaserieselasticactuatorisde- d Velocity Sourced Gearbox T signed to generate a spring deflection force/torque to follow DC Motor o Spring agivencommandtrajectory. Variouscontrolapproacheshave Cable Load beendevelopedfordifferentSEAstoachievehighforce/torque tracking performance. A Proportional-Integral-Differential (PID) control method was introduced in [10] to illustrate the Fig.1: Thevelocitysourcedcable-drivenSEA. concept and performance of a SEA explicitly. A lot of re- searchers have made important contributions to the develop- Neglecting the deflection of the cable, the velocity of the mentofSEAbasedonPIDcontrol[8,9,11–13]. Disturbance cableisgivenas ω observer(DOB)basedcontrolmethodshavebeenalsoadopted ω = . (1) in[14–16]toenhancerobustness. Wyethproposedacascaded m Kg torquecontrolmethodwithaninnervelocitylooptoovercome Where,K isthegearboxratioandωistheoutputvelocityof g problemsofnon-linearities[11]. theDCmotor. Then,thedisplacementofthecableis In this paper, the torque controller is designed using the 2- ω DOF method to track torque reference, eliminate disturbance θm = sm. (2) Ifthedisplacementoftheloadisθ ,thespringdeflectionθ 2.3 DesignoftheVelocityController l s canbederivedas InordertocontroltheDCmotortobeaneffectivevelocity θs =θm−θl. (3) source, a PI controller is designed there. As shown in Fig. 2, ThetorqueT appliedtotheloadisduetothedeflectionofthe thevelocitycontrollerhastheformof o spring,suchthat K To =Ksθs. (4) Cv(s)=Kpv+ siv. (10) Iftheinertiaanddampingofthespringareconsidered, itbe- comes The velocity controller is tuned in the case of no load at- T =(M s2+C s+K )(θ −θ ). (5) tachedandneglectingtheCoulombfriction,thatistosay,the o s s s m l loadtorqueT andtheCoulombfrictionf aresetaszero. To Letθ =0andcombine(1),(2)and(5),thereis l c l create a system that will be of type 1, change (9) and (10) to To(s) = Mss2+Css+Ks. (6) thefollowingform: ω(s) K s g ω(s) Kt 2.2 VelocitySourcedDCMotor G (s)= = JLa , (11) v v (s) (s+p )(s+p ) a 1 2 To design a well performed SEA, a DC motor is used as tahdeopvteeldociistythsaotuvrecleoocfitythceoancttruoaltoisr.eTashieerreaansdonmtohraetstthriasigidhetfaoirs- C (s)= Kpv(s+ KKpivv). (12) v s ward than current control. What’s more, velocity control can overcome some undesirable effects caused by motor internal Wherep2 > p1 > 0andbothcanbeobtainedfrom(9)and disturbance. (11). Let K The schematic diagram of a velocity sourced DC motor is iv =p , (13) K 1 shown in Fig. 2. The velocity feedback regulated by a well pv tuned PI controller forms a stable closed loop so that the DC then,theopenlooptransferfunctionbecomes Motorsystemcantrackthereferencesignalquicklyandaccu- rately. Hopen =Cv(s)Gv(s) ffcc Tl = Kpv(s+ KKpivv) · JKLta s (s+p )(s+p ) (14) d  KpvKsiv vav  Las1Ra ia Kt   1J  1s1s  KpvKt 1 2 b = JLa . K K s(s+p ) b f 2 Theclosedlooptransferfunctionwillbecomeatypicaltwo Fig.2: ThediagramofthevelocitysourcedDCmotor. ordersystem,andhastheformof Whenthecurrenti flowsthroughthemotorcoil,thereis ω2 a H = n . (15) L dia +i R +v =v . (7) closed s2+2ξωns+ωn2 a dt a a b a Accordingto(13),(14)and(15),thereare Wherev istheappliedvoltageacrossthemotorterminals,R a a itshethmeoretosirsctaonilc,eaonfdthvebmisotthoerbwaicnkdienmg,fLvoalitsagtheethinadtuisctlainnecaerolyf  Kpv = ωn2KJtLa proportional to the angular velocity ω of the motor, such that . (16) K =p K vb =Kbω.  iv 1 pv Theequationoftheshaftrotatingaboutafixedaxisis 2ξωn =p2 K i −K ω−f −T =Jα. (8) Given the desired performance of the closed loop system, t a f c l thenaturalfrequencyω anddampingratioξcanbeestimated. Where α is the angular acceleration such that α = ωs. The n Then, the controller parameters can be obtained from (16). magnetictorqueislinearlyproportional(K )tothecurrenti t a Further, the transfer function from the desired velocity input flowing through the motor coil. The viscous friction torque ω (s) to the actuator output torque T (s) can be obtained by is in the opposite direction of the motion and is linearly pro- d o T (s) portional (Kf) to the angular velocity ω. fc is the Coulomb P(s)= o . ω (s) frictionandT istheloadtorque. d l Substitute (7) into (8) and convert it to frequency domain 3 TorqueControllerDesign equation,ω(s)canbeobtainedas 3.1 ControllerwithOptimalTransient v (s)K −(L s+R )(f (s)+T (s)) ω(s)= a t a a c l . (9) In this subsection, a possible way to design the best stabi- JL s2+(JR +K L )s+K R +K K a a f a f a b t lizing controller and measure the quality will be introduced. AssumethatP(s)isstrictlyproperanddenotedastheformof 3.2 Stabilizing2-DOFController b(s) b sn−1+···+b Aclosedloopsystemusing2-DOFcontrollertostabilizethe P(s)= = 1 n . (17) a(s) a sn+a sn−1+···+a plantP(s)isshowninFig.4. Theoutputzcantracktherefer- 0 1 n enceinputsignalr inasatisfactoryway,also,thedisturbance Wherea(s)andb(s)arecoprimeanda (cid:54)=0. 0 d and the noise n can be eliminated close to zero in a short time. w u 1 1 P(s) d  y y r u v z 1 2 C(s) P(s) u w C(s) 2 2  y n Fig.3: Feedbacksystemforstabilization. Fig.4: The2-DOFcontrolconfiguration. ConsiderthesystemshowninFig.3,iftheclosedloopsys- temisinternallystableandtheexternalinputsw (t),i = 1,2, i The stabilizing 2-DOF controller based on the controller areallimpulsesignals,alltheinternalsignalsu (t),y (t),j = j j withoptimaltransientisshowninFig.5. LetC (s)beasta- 0 1,2, will eventually settle to zero when time goes to infinity. q(s) So,theRMSvalueofyi(t)i = 1,2,whenwi(t),i = 1,2,are bilizingcontrollerwithoptimaltransient: C0(s) = p(s). Let unitimpulsescanbeusedtomeasurethequalityoftheperfor- M(s),N(s),X(s),Y(s)be: manceasfollow: a(s) b(s) J =(||y1(t)||22+||y2(t)||22)|w1(t)=δ(t) M(s)= d(s),N(s)= d(s), w2(t)=0 (18) (21) +(||y (t)||2+||y (t)||2)| . p(s) q(s) 1 2 2 2 w1(t)=0 X(s)= ,Y(s)= . w2(t)=δ(t) d(s) d(s) For different stabilizing controllers, their RMS value J are Then M(s),N(s),X(s),Y(s) are all stable transfer func- different. TheminimumvalueofJ denotedbyJ∗ whenC(s) tionssatisfying: ischosenamongallstabilizingcontrollersmeansthatthebest performanceisachieved. N(s) Y(s) P(s)= ,C (s)= , Thetransferfunctionfrom(w1,w2)to(y1,y2)is: M(s) 0 X(s)  P(s)C(s) C(s)  M(s)X(s)+N(s)Y(s)=1. 1+P(s)C(s) 1+P(s)C(s)    . (19)  P(s) −P(s)C(s)  d 1+P(s)C(s) 1+P(s)C(s) r u v z Q(s) X1(s) P(s) 1 The next problem to be solved is how to find the most optimal stabilizing controller. Firstly, there must exist a stable polynomial d(s) called spectral factor of a(−s)a(s) + N(s) b(−s)b(s)suchthat: a(−s)a(s)+b(−s)b(s)=d(−s)d(s). q(s) Q (s) Now let C(s) = be the unique n-th order strictly 2 p(s)   properpoleplacementcontrollersuchthat: M(s) a(s)p(s)+b(s)q(s)=d2(s). Then, this controller C(s) is the most optimal controller Y(s) y n whichminimizesJ andgives d(s)−a(s) b(s) Fig.5: Stabilizing2-DOFcontrolstructure. J∗ =|| ||2+|| ||2 d(s) 2 d(s) 2 (20) d(s)−p(s) q(s) Denotethesetofallthe2-DOFcontrollerswhichgivestable +|| ||2+|| ||2. d(s) 2 d(s) 2 closedloopsystemsbyΩ(P): (cid:20) (cid:21) If w ,i = 1,2, are external disturbances or noises, a stabi- Q (s) Y(s)+M(s)Q (s) i Ω(P)={C(s)= 1 2 }. lizingcontrollerwithoptimaltransientcaneliminatethemina X(s)−N(s)Q (s) X(s)−N(s)Q (s) 2 2 shorttime. (22) Where,Q (s),Q (s)arearbitrarystabletransferfunctions. tobe50Hz,let 1 2 Every stabilizing 2-DOF controller has the form shown in 1 Fig.5. ForafixedplantP(s),ifweplugastabilizing2-DOF Q (s)= . 2 s/(100π)+1 controllerofthisformtoaclosedlooptransferfunction,then theclosedlooptransferfunctionbecomesafunctionofQ1(s) 4 SimulationsandResults and Q (s). The problem becomes choosing good Q (s) and 2 1 Thecable-drivenSEAparametersarelistedinTable1. Q (s) to meet the design specifications, as long as they are 2 stable. Table1: Parametersofthecable-drivenSEA. The four transfer functions from r to the internal variables u,v,y,zdependonlyonQ1(s): J 6.96×10−6kg·m2 L 0.62mH  C (s)  a 1 R 2.07Ω a  1+P(s)C2(s)  K 0.0525Nm/A   t  C (s)    K 0.0525Vs/rad  1  M(s) b  1+P(s)C2(s)   M(s)  Kf 0.00001Nm/(rad/s)  P(s)C (s) = N(s) Q1(s). (23) Ks 138Nm/rad  1+P(s)1C2(s)  N(s) KCsg 0.01N1m56/:(r1ad/s)  P(s)C1(s)  Jl 0.1kg·m2 M 0.00001kg·m2 1+P(s)C (s) s 2 The eight transfer functions from d,n to u,v,y,z depend onlyonQ2(s): 4.1 SimulationResultsoftheVelocityControl  −P(s)C2(s) −C2(s)  Thetransferfunctionfromva(s)toω(s)is:  1+P(s)C2(s) 1+P(s)C2(s)  v (s) 1.217e07   a = .  1 −C2(s)  ω(s) s2+3340s+6.435e05  1+P(s)C (s) 1+P(s)C (s)   2 2   = When the damping ratio ξ is chosen to be 0.88, the co-  P(s) 1    efficients of the proportional-integral controller are Kpv =  1+P(s)C2(s) 1+P(s)C2(s)  0.26,Kiv = 53.5. The step response of velocity closed loop    P(s) −P(s)C (s)  systemisshowninFig.6. Therisetimeofthevelocityclosed 2 1+P(s)C (s) 1+P(s)C (s) . (24) loop system is within 0.002s, almost no overshoot, can meet 2 2  −N(s)Y(s) −M(s)Y(s)  therequirementsofavelocitysource.  M(s)X(s) −M(s)Y(s)     N(s)X(s) M(s)Y(s)  N(s)X(s) −N(s)X(s)   M(s)  M(s)  (cid:2) (cid:3) − N(s) Q2(s) N(s) M(s) N(s) ThismakeschoosingQ (s)andchoosingQ (s)decoupled 1 2 andratherconvenient. 3.3 ParameterizationofQ (s)andQ (s) 1 2 Sincethetransferfunctionfromthereferenceinputrtothe Fig.6: Stepresponseofthevelocitycontrol outputsignalzisN(s)Q (s),let 1 ω¯2 Q1(s)= N(s)(s2+2nξ¯ω¯ s+ω¯2). 4.2 SimulationResultsoftheTorqueControl n n The transfer function P(s) from ω (s) to T (s) when the d o Then,thetransferfunctionfromrtozbecomes loadisfixedis: N(s)Q1(s)= s2+2ξ¯ω¯ω¯n2ns+ω¯n2. P(s)= 0.20s344+s33+34204s53.+1s32.+8127.e80468se20+6s6+.545e.70681se08 Considering that Q (s) is mainly used to filter the distur- Thestabilizing2-DOFcontrollercanbeobtainedbychoos- 2 bancedandnoisenwhoseindustrialfrequenciesareassumed ing ω¯ = 451.24 and ξ¯ = 0.826. In the simulation, a PD n d a cable-driven SEA. The 2-DOF torque controller performed T u  T bettercomparedwiththePDmethodinthepresenceofnoise d C(s) d P(s) o anddisturbanceinthesimulations. Thetorquecontrolperfor- mance can be conveniently adjusted by choosing appropriate  filterQ (s)andQ (s). 1 2 Further research will be focused on the impedance control y n ofthecable-drivenSEA. Fig.7: Torquecontrolstructureforthecable-drivenSEA. References [1] G.A.PrattandM.M.Williamson. Serieselasticactuators. In Proceedings of the IEEE/RSJ International Conference on In- controllerC (s) = 490+0.1sisusedforcomparison. The PD telligentRobotsandSystems,volume1,pages399–406,1995. torque control structure for the cable-driven SEA is shown in [2] D.W.Robinson,J.E.Pratt,D.J.Paluska,andG.Pratt. Series Fig.7. elastic actuator development for a biomimetic walking robot. 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