Control And Evaluation Of Series Elastic Actuators With .
Control and Evaluation of Series Elastic Actuators with NonlinearRubber SpringsJessica Austin1 , Alexander Schepelmann1 , and Hartmut Geyer1Abstract— Series elastic actuators primarily use linearsprings in their drivetrains, which introduces a design tradeoff:soft springs provide higher torque resolution at the cost ofsystem bandwidth, whereas stiff springs provide a fast responsebut lower torque resolution. Nonlinear springs (NLSs) potentially incorporate the benefits of both soft and stiff springs,but such springs are often large. An NLS design was recentlyproposed that combines a variable radius cam with a rubberelastic element, enabling a compact spring design. However,the rubber introduces hysteresis, which can lead to poortorque tracking if not accounted for in the controller. Toovercome this limitation, we here propose a state observer thatcaptures hysteretic effects exhibited by the rubber to providean accurate estimate of actuator torque. We perform torquecontrol experiments with this observer on an actuator testbedand compare the performance of the NLS to both soft and stifflinear metal springs. Experiments show that the NLS exhibitsimproved output impedance compared to both linear springs,and comparable bandwidth to the stiff linear spring up to 1.5Hz. However, the hysteresis in the urethane rubber introducesinstability in higher-frequency conditions, suggesting that futureNLS designs can be improved by use of a different rubber asthe elastic element.I. I NTRODUCTIONSeries elastic actuators (SEAs) provide many benefitsthat make them attractive for legged robotics and poweredprostheses, including greater shock tolerance, low outputimpedance, passive energy storage, and more accurate forcecontrol [1] [2] [3]. Traditionally, SEAs use linear springs—typically metal—as their elastic element, as these are cheap,widely available, and follow Hooke’s Law for force estimates. However, the use of linear springs introduces adesign tradeoff: in a system with set encoder resolution, softsprings provide higher torque resolution at the cost of systembandwidth, whereas stiff springs provide a fast response butlower torque resolution [4].Variable stiffness actuators attempt to overcome this tradeoff by achieving a range of stiffnesses via actively tuninga passive mechanical element with a secondary motor [5][6] [7]. However, these systems are often bulky or complicated, hampering their application in small, lightweightrobots, or for retrofit in existing robots. Passive nonlinearsprings (NLSs) reduce complexity by omitting active tuningof the mechanical element, and encode a single nonlineartorque profile [8] [9]. To further reduce weight and volume,rubber is an attractive alternative to metal for these springs’*This work is supported by the Eunice Kennedy Shriver National Instituteof Child Health & Human Development under award no. 1R01HD075492.1 J. Austin, A. Schepelmann, and H. Geyer are with the RoboticsInstitute, Carnegie Mellon University, Pittsburgh, PA, 15213, USA.{jaustin,aschepelmann,hgeyer}@cmu.eduelastic element because it is compact, can tolerate largedeflections, and can be easily molded to a custom shapeand size. For these reasons, rubber has been incorporatedinto actuator designs where weight and volume is a concern[10]. Compared to metal springs, viscoelastic materials likerubber have the disadvantage of exhibiting hysteresis due toviscous effects, though others have overcome this challengeby using state observers to account for hysteresis [11].To improve bandwidth and torque resolution in an SEAwhile maintaining a lightweight, compact design, we previously developed an NLS design that combines a variable radius cam with rubber springs (Fig. 2) [12]. This design allowsfor an arbitrary, user-defined torque-deflection profile andstrives to offer the improved torque tracking of soft springsat low amplitudes and the fast response of stiff springsat high amplitudes. However, the same rubber that allowsfor a compact design also exhibits a nonlinear stress-strainprofile and hysteretic effects (Fig. 1). If unaccounted for,these behaviors degrade the overall stability and bandwidthof closed-loop torque control, and because they are nonlinearand time-dependent, simple approaches such as Hooke’s Lawor a lookup table are infeasible. Therefore, in addition tochoosing a rubber that minimizes undesirable behaviors, thedesigner must also develop a more sophisticated model toprovide an estimate of rubber state in the controller.In this work, we focus on development of a rubbermodel and corresponding state observer for a single typeof urethane rubber; this observer captures nonlinear andhysteretic effects exhibited by the rubber in order to providean accurate estimate of actuator torque. We perform hardwareexperiments on an actuator testbed to evaluate both the stateobserver and the performance of the overall system. Ourrubber model has an average percent relative error of 13.6%in experiment, which is a significant improvement over theerror in a Hooke’s Law fit of 25.2%. The resulting observer isstable and gives estimates as accurate as its internal rubbermodel. Placing this observer in closed-loop with a torquecontroller, we compare the performance of our NLS to bothsoft and stiff linear springs. Experiments with the NLS showimproved output impedance compared to both linear springs,and comparable bandwidth to the stiff linear spring up to 1.5Hz. However, hysteresis in the rubber introduces instabilityin higher-frequency conditions; therefore, careful selectionof an alternative rubber is recommended for future designs.II. N ONLINEAR S PRING SEA D ESIGNThe nonlinear spring design described in [12] is a two-partassembly, consisting of an elastic element and a rotary cam
Mean torque (Nm)2.5Top View20.17 Nm1.5Cable(17%)Rubber10.5High AmplitudeLow Amplitude000.10.20.30.40.50.6Cam rotation (rad)Frτ rubber3.4cm3cmRubberβCableRubberCamCableLo 1.25cm(a)CamLoad cellCableCam ProfileCable4:10.7Fig. 1: Average torque profile for prototype NLS SEA. Averagesare based on experimental data using open-loop position control,with velocities ranging from 0.1–9 Hz, and for low-amplitude( θ 25 ; ncycles 354) and high-amplitude ( θ 45 ;ncycles 222) scenarios. A nonlinear torque profile results fromboth the nonlinear cam design and the rubber behavior under strain.Hysteresis on downstroke comes from using rubber as the spring.RubberFront ViewRE 40Cam Profile(b)Fig. 2: NLS SEA cam prototype components (a) and schematic (b).A cable runs through the center of the cam and attaches to rubbersprings at either end. As the cam rotates, the rubber stretches and thecable engages the cam at different points on the profile, effectivelychanging the lever arm r. The SEA torque, τrubber , is the crossproduct of the lever arm r and the rubber force vector F.Fig. 3: Benchtop setup for rubber characterization and observertesting. To simplify experiments, the load side of the rubber is fixed.Load cells in-line with the rubber give rubber force measurementsfor testing, but will not be present in the final, compact design.muscle bandwidth [14], dynamically scaled to match therobotic leg size and mass.Experiments are conducted in a testbed (Fig. 3) that incorporates piezoresistive pressure sensors (FlexiForce A201:Tekscan) to measure the tension in the rubber. These loadcells assist in observer development and will not be presentin the final, compact design. Previous work developed ahigh-fidelity simulation for actuators in the robotic leg; thissimulation runs in MATLAB Simulink with SimMechanicsand includes friction, noise, and encoder discretization [15].The simulation is updated here for use with NLS SEAs, andis used for controller development and to simulate scenariosnot possible in the hardware testbed.Hardware evaluations with the NLS SEA prototypeshowed that the actuator matched the desired torque profileon upstroke at low speeds of 0.1 Hz. However, due tovelocity- and time-dependent effects in the rubber, therewas significant deviation from the desired profile duringdownstroke or at higher speeds.III. A PPROACHwhose profile is optimized to stretch the elastic element overa variable radius (Fig. 2). As the cam rotates, subsequentpoints on the cam profile are engaged, which changes theinstantaneous lever arm and thus the torque generated by thespring. Through choice of the cam profile and choice of therubber, the user can design an NLS with an arbitrary torquedeflection profile, within the constraints of manufacturingtolerances. Motivation for the NLS SEA comes from bandwidth and torque resolution limitations in existing SEAs ina robotic neuromuscular leg testbed [13] that was developedto investigate human neuromuscular controllers.To evaluate the NLS SEA design, a prototype actuator wasdeveloped for incorporation into the robotic neuromuscularleg testbed [12]. Elastic elements are urethane rubber springs(PMC-770, Smooth-On Inc.) 3 cm wide, 3.4 mm thick and1.25 cm long, that are held within a clamp and attached tothe cam via a cable. The cam and rubber together encode anexponential torque profile with a maximum torque of 5 Nm at53 maximum rotation. The system is designed for operatingfrequencies up to 9.8 Hz, which is the upper limit of humanA. Rubber Model DevelopmentIn order to account for nonlinearities and hysteresis inthe rubber, the NLS SEA state-space observer design mustincorporate a model of the rubber that accurately predictsthese effects. Our goal is to model the rubber force; withthis we can use known cam geometry to directly calculatespring torque. Our primary requirement for this model isthe ability to represent behaviors seen in experiments, suchas creep, recovery, and stress relaxation (Fig. 4). Creep isincreasing deflection under a constant force, recovery is anon-instantaneous return to the rest length after the forceis removed, and stress relaxation is decreasing force undera constant deflection [16]. Secondly, since we wish to usethis model for a state-space observer to estimate force, themodel must be linear in force and deflection. Finally, sincewe use this model on an actual system with noisy, discretizedoutputs, we exclude models with high-order derivatives onthese states.A set of rubber models exists that satisfy these criteria:linear viscoelastic models [17]. These models represent
Fepcrerecoverytk1 k2 k1η1η2k2(e)Fig. 5: Candidate viscoelastic models. Springs and dashpots represent elastic and viscous elements, respectively. (a) Hooke’s Law.(b) Kelvin-Voigt model. (c) Maxwell model. (d) Standard LinearSolid model. (e) Burger’s model.rubber, a viscoelastic material, as a mechanical systemcomposed of springs as the elastic elements and dashpotsas the viscous elements (Fig. 5). These elements may beplaced in series or parallel to encode various behaviors. Ingeneral, a model with a greater number of elements offersincreased modeling accuracy, at the expense of mathematicalcomplexity and higher order derivatives. For example, theKelvin-Voigt model encodes creep but not stress relaxation,and the Maxwell model encodes stress relaxation but notcreep; combining these models gives the Standard LinearSolid model and Burgers model, which encode both creepand stress relaxation [17]. However, the Burgers model hashigher-order derivatives on force and deflection, resultingin amplified noise on the hardware system. Therefore wechoose the Standard Linear Solid model as the simplestmodel that encodes our desired behaviors.The constitutive equation for the Standard Linear Solidmodel is given byF k1100RubberLinear fittkη ηk100k1 k2Fig. 4: Rubber behaviors that we seek to model. Left: Creep,which is deflection under a constant force, and stress relaxation,which is decreasing force under a constant deflection. Right: Straindependent stress, which can be approximated by linear stiffnesses.k150k 1 k 2 A0ηA0δ ηδ̇ ḞL0k2 L0k2(1)where k1 and k2 are stiffnesses, η is the viscosity, F is therubber force, and δ is the change in length of the rubber.Note that rubber models are typically given in terms ofstress, σ, and strain, ε; in this paper, we give all equationsin terms of F and δ. The conversion is made using σ AF0and ε Lδ0 , where A0 is the cross-sectional area of therubber and L0 is the rest length. Each of the three termsof equation (1) contributes to the desired rubber behavior:the first term is essentially Hooke’s Law and contributes toelasticity, the second term contributes to creep, and the third0Exp. (High)Model (High)Exp. (Low)Model (Low)5050k1t1150F (N)t1tk1 k2 k3 k4F (N)t1δFδF10.6410.6610.68t (sec)10.7010.720012δ (mm)34Fig. 6: Sample fit from Std. Lin. SDS model characterization,which incorporates creep, stress relaxation, and strain-dependentstiffnesses. Experimental data are from a single cycle at 4 Hz, forboth low and high amplitudes. Left: Force vs time. Right: Forcevs change in length of the rubber. The jagged lines are a result offinite encoder resolution.term contributes to stress relaxation [17].Experimental data also reveal that our rubber exhibitsstrain-dependent stiffness (Fig. 4), which the Standard LinearSolid model, with its constant stiffnesses k1 and k2 , is unableto model. However, we can approximate the nonlinear forcedeflection curve with piecewise linear stiffnesses; the forceequation for such a spring can be written asF nA0 X k Hδ δi (δ δi )L0 i i(2)where Hδ δi is the Heaviside step function centered at δi .The stiffness of k changes based on the amount of deflectionin the rubber; it behaves as n springs in parallel, each ofwhich “engages” once the rubber reaches a certain deflection.Replacing spring k2 in the original model with this nonlinearspring and deriving the equations of motion giveskηnA0 X k Hδ δi (δ δi )L0 i i!nXηη A0 ki δ̇ Ḟk k L0kiF k*(3)which we call the Standard Linear Solid model with StrainDependent Stiffness (Std. Lin. SDS). This model allows usto model strain-dependent stresses while keeping the modellinear in δ for any particular δ, as will be shown in thefollowing section.To fit model parameters, we conduct characterization experiments, split into training and testing datasets, and runstochastic optimization [18] to minimize average percentrelative error between the model and the actual data. Inthese experiments, we drive the SEA cam in sinusoidsat amplitudes of 15 , 25 , and 45 , and at frequenciesranging from 0.1-9 Hz. Cam position, motor current, andforce are measured. The value of δ is calculated based onthe cam rotation θ along with known cam geometry, andF is measured directly using the tension load cells. Thederivatives δ̇ and Ḟ are calculated using a bi-directional lowpass filter.The parameters for the Std. Lin SDS model are: A0 1cm2 , L0 1.25 cm, η 411 Pa·s, k 0.58 MPa, δ1 0 mm,
k1 0.92 MPa, δ2 0.20 mm, k2 0.58 MPa, δ3 1.06mm, k3 0.74 MPa, δ4 2.65 mm, k4 0.77 MPa, δ5 9.04mm, k5 0.47 MPa, δ6 9.05 mm, and k6 1.06 MPa. Arepresentative fit against one cycle of characterization datais given in figure 6. Against the full characterization dataset, the model achieves an average percent relative error of13.6%, which is a significant improvement over the error in aHooke’s Law fit of 25.2%. This fit corresponds to an RMSEof 0.10 Nm at the spring. To put this in context, we use thesame SEA setup fitted with a linear spring of comparablestiffness to the NLS at small rotations (that is, higher torqueresolution), and use Hooke’s Law to estimate spring stiffness;this gives a fit with RMSE of 0.07 Nm, which comes fromencoder discretization and noise in the load cells. Therefore,the developed rubber model approaches the limit of what canbe achieved with imperfect sensing.B. State-Space EquationsOur rubber model (3) relates force, F , and change inlength of the rubber, δ. In order to improve the estimateof F , we can also take advantage of the motor dynamics,which relate the known torque at the motor, τmotor , to theunknown rubber torque, τrubber ,J θ̈ τmotor τrubber nkT I rF sin β,(4)where I is the motor current, kT is the torque constant, andn is the gear reduction between the motor and the spring.Putting this together with the rubber model equation (3), andδ δ(θ), gives state-space equations for the cam:ẋ A(θ)x Bu E(θ)y Cxwhere ka12 00Fηdδ δ 00 0 dθ(θ) x θ , A(θ) 00 01 r sin(β(θ))θ̇0 00 J P6A00k i 1 ki L0 δ̇ 00B , u I 00 kT n0 J 6Pk A0ki Hδ(θ) δi δi η L0 i 1 , C 0 1 0 0E(θ) 0 0 0 1 0 00P6 0and a12 kη Ai 1 ki Hδ(θ) δi . Note that E(θ) is simL0ply a constant offset matrix required to linearly interpolatebetween step changes in the force for the spring k .Since β and δ change with the cam position, θ, the Aand E matrices are not time-invariant. However, they can becalculated for all the possible values of θ, so in the observerpreprocessing code a lookup table is created for A(θ) andE(θ) for θ ( π2 , π2 ). τdesPIDτobsθdesObsSEAτactualLoadcellsθ,IFig. 7: Plant used for closed-loop system identification. The observer provides an estimate of spring torque, which is subtractedfrom a desired torque and fed into a PID controller. The controllerconverts this error to a velocity command that is sent to the motorcontroller, and sensors provide estimates of θ and motor current I.C. Observer DesignIn our system, we implement a Luenberger Observer [19],x̂ A(θ)x̂ Bu L(y C x̂) E(θ)(5)where we select values for the observer gain matrix L usingpole placement; the poles are chosen via optimization tominimize the error between state estimates and actual datagathered during characterization.To check for T the observability observability, we calculatefor each value ofmatrix N C CA CA2 CA3θ. In most cases, N is full rank, but when β π it is rank3 and thus the system is unobservable. Pole placement inunobservable conditions is impossible, and this occurs whenthe cam is passing through θ 0 and so is unavoidableduring typical use. However, using Kalman decomposition[19] we can decompose the system into observable and unobservable sub-systems, Aobs , Cobs and Aunobs , Cunobs .The observable sub-system is composed of the states F , δ,and θ̇. The unobservable sub-system is the state θ and haseigenvalue less than zero, so it is asymptotically stable. Sincethe unobservable sub-system is stable, we can determine Lusing pole-placement with the following algorithm: whenθ 6 0, choose L to satisfy desired closed-loop poles forA LC, and when θ 0, perform pole placement forthe observable sub-system Aobs LCobs while leaving theunobservable sub-system poles at zero.Since our state-space matrices and observer gain matrixchange based on the cam rotation θ, our design implements aform of gain scheduling. With the exception of some specialcases, there is no method to prove global stability for asystem with gain scheduling—instead the researcher mustverify stability through experiment [20]. In our case, thedeveloped observer is indeed stable, both in simulation andon the hardware testbed. To characterize the observer error,we run simulations and compare against characterizationdata; the observer achieves 13.7% average percent relativeerror and 0.10 Nm RMSE—the same performance as therubber model.IV. P ERFORMANCE A SSESSMENTThe original goal of the NLS SEA was to provide thetorque resolution of a soft linear spring at low amplitudes andthe bandwidth of a stiff linear spring at high amplitudes, all ina lightweight and compact design. To evaluate whether the
τdes τactual5Torque (Nm)2.8NmMagnitude (dB)43.4 Nm/rad0.8Nm1.7 Nm/rad020-2-3dB-4-60-5-0.8 -0.6-0.4-0.200.20.40.6Phase (deg)NLSSoft linear springStiff linear spring0.8-50Stiff Metal-100Soft MetalNLSCam Rotation (rad)01010010Frequency (Hz)14Magnitude (dB)Fig. 8: Average experimental torque profiles for each spring, withstandard deviations. Dashed lines indicate extrapolated data
Control and Evaluation of Series Elastic Actuators with Nonlinear Rubber Springs Jessica Austin 1, Alexander Schepelmann , and Hartmut Geyer Abstract Series elastic actuators primarily use linear springs in their drivetrains, which introduces a design tradeoff: sof
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Section 2 Evaluation Essentials covers the nuts and bolts of 'how to do' evaluation including evaluation stages, evaluation questions, and a range of evaluation methods. Section 3 Evaluation Frameworks and Logic Models introduces logic models and how these form an integral part of the approach to planning and evaluation. It also
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3 Evaluation reference group: The evaluation commissioner and evaluation manager should consider establishing an evaluation reference group made up of key partners and stakeholders who can support the evaluation and give comments and direction at key stages in the evaluation process.
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