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A fluidic-muscle driven force-controlled parallel platform for physical simulation of virtual spatial force-displacement laws

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A fluidic-muscle driven force-controlled parallel platform for physical simulation of virtual spatial force-displacement laws
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  Meccanica (2011) 46: 171–182DOI 10.1007/s11012-010-9407-8 PARALLEL MANIPULATORS A fluidic-muscle driven force-controlled parallel platformfor physical simulation of virtual spatial force-displacementlaws Mahendra Dhanu Singh  · Kusnadi Liem  · Vladimirs Leontjievs  · Andrés Kecskeméthy Received: 30 September 2009 / Accepted: 6 December 2010 / Published online: 4 January 2011© Springer Science+Business Media B.V. 2010 Abstract  Described in this paper is a six-leggedStewart-Gough parallel platform driven by a relativelynew type of fluidic muscles. The advantage of the plat-form is that it is virtually free of stick-slip effects.Thus, the device is well-suited for fine-tuned forcecontrol and for physical simulation of virtual force-displacement laws. The legs of the platform are of type RRPS and are equipped with a coaxial coil springand a fluidic muscle providing push and pull forces.Each leg is equipped with a force sensor, a pressuresensor, and a magnetostrictive position encoder. Thecontrol for the platform consists of six control loopsfor the six operated actuators with model-based forcecontrol comprising individual gas models as well asthe rubber nonlinearities for each leg. The control lawalso includes the gas flow in the proportional direc-tional control valve in 3/3-way function. The presentpaper describes the basic architecture of the platform,the dynamic models, as well as testbed results for theexisting fluidic-muscle parallel platform DynaHex. Itis shown that the presented control scheme leads to astableforcecontroloftheplatformforquasi-staticmo-tion. As an application, the device will be employed infields of biomechanics, as well as in general environ-ments requiring physical simulation. M. Dhanu Singh  ·  K. Liem  ·  V. Leontjievs  · A. Kecskeméthy (  )Lehrstuhl für Mechanik und Robotik, UniversitätDuisburg-Essen, 47057 Duisburg, Germanye-mail: andres.kecskemethy@uni-due.de Keywords  Fluidic muscle  ·  Force control  ·  Parallelplatform  ·  Physical simulator 1 Introduction Parallel platforms are well-established today and areavailable in multiple architectures and technologies[1]. In this setting, most parallel platforms are positioncontrolled [2]. A special kind of task results when theparallel manipulator is targeted to provide force con-trol. In these cases, one seeks to control the actuatorssuch that a given force at the end-effector is achieved.Current methodologies for this task are based on hy-brid force/position control schemes, where the actua-tors are velocity-controlled with some assumptions onthe compliance of the environment [3]. For example,the force-impedance control scheme proposed in [4]uses a robot system that consists of an industrial serialsix-axis robot and a small six DOF parallel manipula-tor fixed at its flange. By this approach, the merits of the large workspace of the serial robot are combinedwiththehighbandwidthoftheparallelplatformdrivenbyelectricmotors.Thisconceptisalsousede.g.forin-dustrial force-feedback control [5]. However, in someapplications, such as very stiff environments, or in en-vironments with unknown or discontinuous stiffnessproperties, direct force control would be more desir-able. An example of such an application is a testingdevice for cervical pairs [6], where sudden facet jointcontact would lead to probe damage due to errors in  172 Meccanica (2011) 46: 171–182 position control. An approach to circumvent this prob-lem is the use of fluidic muscles, which have the ad-vantage of avoidance of slip-stick effects, and whichalso have a very good transmission behavior betweenpressure and force. Additional advantages of fluidicmuscles are their small size with respect to achiev-able forces, as well as their long durability. Pneumaticmuscles are already used to control the pose of paral-lel platforms [7]. By active force control, such fluidicmuscles can produce any desired force-displacementlaw. In this paper, we analyze the use of pneumaticmuscles for directly controlling the force at the plat-form. Although fluidic muscles are more difficult tocontrol, we make use of the compliance and directforce controllability to improve the end-effector be-havior when sudden contact arises. The optimal designof the platform has to take into account the problemsof singularity avoidance [8], optimization of manipu-lability index [9] and dexterity index [10], direct kine- matics computation [11], as well as general technolog-ical issues [1].This design task is described in the first part of thepaper, the modeling and control of the pneumatic mus-cles is described in the second part and finally, experi-mental results obtained with the developed force con-troller are presented in the third part. 2 Platform design 2.1 Basic structureFor the realization of the targeted platform, a six-legged  RRPS  -type parallel platform [1] was chosen,each leg being hinged by a universal joint (RR) at thebase and a spherical joint (S) at the platform, witha prismatic joint along its leg axis as input stroke(Fig. 1).In order to avoid slip-stick effects, the actuators arechosen as fluidic (pneumatic) muscles (MAS 20, FestoAG & Co. KG. [12]), which operate by contracting inaxial direction when inflated.2.2 ActuatorsAs afluidicmusclecanonlyproducepullingforces,anadditional force element is required in order to providealsoapushforce.Thisisrealizedinthepresentcontextby embedding the fluidic muscle into a coaxial coil Fig. 1  Concept of the developed parallel manipulator Fig. 2  Single actuator with force, position and pressure sensors spring whose stiffness is determined such that the re-quired maximal pushing forces can be obtained withinthe stroke of the fluidic muscle (Fig. 2). Each actua-tor is equipped with sensors measuring force, pressureand stroke, as illustrated in Fig. 2. The stroke of thehybrid actuator depends on the specific type of the flu-idic muscle and its length. For the chosen fluidic mus-cle MAS 20, the stroke is  ℓ =  40 mm.The relationship between air pressure and contrac-tionforceisnonlinearanddependsontheactuallengthof the muscles (Fig. 3).The maximal and minimal actuator forces  F  min , F  max  depend on the current length  ℓ  of the actuatordue to the nonlinear characteristic of the fluidic mus-cle: at a constant pressure the muscle force decreasesif the contraction increases. The corresponding maxi-mal and minimal forces in Newton [N] at the attach-ment point of the platform (“ + ”  =  pull, “ − ”  =  push)are plotted for the current actuators in Fig. 4. The cor-responding relationships can be approximated by the  Meccanica (2011) 46: 171–182 173 Fig. 3  Force-contraction curves of fluidic muscle Fig. 4  Maximal and minimal actuator forces as function of ac-tuator length equations F  min  =  7Nmm ℓ − 4 , 768 . 5 N F  max  =  0 . 304Nmm 2 ℓ 2 − 361 . 3Nmm ℓ + 107 , 095 . 7 N2.3 Platform parametersAs basic design concept for the platform, a  symmet-ric simplified manipulator   [13] was chosen, where theattachment points at the base and the platform are lo-cated along a circle in three symmetrically distributedpairs of attachment points, respectively (Fig. 5). Thusone needs to consider as design parameters only therelative angle offsets  α A  and  α B  and the circle radii  r a and  r b  at base and platform. This has many advantagesin manufacturing.As an additional design parameter, the height  h EE of the end-effector location with respect to the planepassing through the platform spherical joints (Fig. 6A)was considered. As will be seen, this parameter hassignificant influence on the required stroke of the ac-tuators for completing a given platform motion.A further important issue was the manufacturing of the spherical joints, which needed to feature tilt anglesof   ± 25 ◦ in order to allow for the desired large roll andpitch angles of the platform. For this purpose, a cus-tomized version was built consisting of a two-piecesocket manufactured from brass and held together byspring screws (Fig. 6B).For the placement of the hinge points at base andplatform, a theoretical investigation of the best place-ment points was conducted in [6].The design consisted of the phases: (1) finding asingularity-free basic configuration, and (2) verify-ing by interval analysis that the platform will achievethe expected results (stiffness, collision-avoidance,limited stroke, expected forces) within the targetedworkspace. Fig. 5  Attachment points of the joint centers  174 Meccanica (2011) 46: 171–182 Fig. 6  ( A ) Distance of the end effector from the platform plane. ( B ) Limit of spherical joint In order to achieve a singularity-free design of theplatform, the determinant of the inverse of the end-effector Jacobian  J − 1 EE  should greater than zero, whichcorresponds to a positive “manipulability index” [9].The end-effector Jacobian  J EE  hereby maps infinites-imal length changes  δ ℓ  of the legs to correspondinginfinitesimal variations  δt  EE  of the platform pose atthe end effector: δt  EE  =  J EE δ ℓ .  (1)For the given architecture, it is easier to computethe inverse  J − 1 EE  of the Jacobian. This can be donee.g. using the concept of kinematical differentials [14],which is not reproduced here as the correspondingequations are trivial.A possible, plausible criterion for the design of theplatformistominimizetherequiredstroketomovetheplatform, such as to have large platform motions withrestricted fluidic muscle strokes. To this end, first, thestrokeasafunctionofangle α A  wasregarded,showingthat the minimal stroke is attained for  α A  = α B , andin particular for  α A  =  0 ◦ . However, the manipulabilityindex at this point always has a minimum equal to zerofor  α A  = α B , meaning that a configuration  α A  = α B  isalways singular. Hence, this optimizing criterion is notsuitable for the platform design.As a second possibility, the dependency of the legstroke on the end-effector offset was regarded. Thisis shown in Fig. 7, where the stroke is plotted overthe upper platform radius for different values of theplatform offset. As can be seen, an operation point be-low of the platform always leads to higher requiredstroke, while positive offset values reduce the amountof the required stroke. Taking  h EE  =  125 mm, onecan obtain a stroke of 40 mm at the prescribed up-per radius of 100 mm, which is a realistic stroke for Fig. 7  Strokeasfunctionof  h EE  and r B  ( α A  = − 20 ◦ ,α B  =  5 ◦ ) the chosen fluidic muscles. This shows that choosinga suitable operation point is also relevant for an ap-propriate design of the platform. For the chosen an-gle values of   α A  = − 20 ◦ and  α B  =  5 ◦ , the platformwas proven to be singularity-free in the whole targetedworkspace [6].In addition to singularity avoidance, the rangeof motion and the target forces of the end-effectorwere analyzed in [6] using the interval-arithmeticlibrary ALIAS [15]. Hereby, the functionality wasverified throughout the complete workspace of theplatform defined by the maximal stroke interval ℓ min  =  625 . 5 mm and  ℓ max  =  665 . 5 mm. 3 Modeling and control of fluidic muscles 3.1 Model a fluidic muscleFor developing a model-based control, the characteris-tics of each fluidic muscle were identified by individ-  Meccanica (2011) 46: 171–182 175 ualmeasurementsduetopossibledifferencesinthesixactuators. At first, the relationship between force  F   onthe one side, and stroke  s  and pressure  p  on the other(as depicted by Fig. 4) was approximated by a ninthorder polynomial: F   = g(s,p) ≈ i + j  ≤ 9  i,j  = 0 a ij  s i p j  (2)The coefficients were obtained from experimentsanddatafittingusingleastsquares[16].Thehighpoly-nomial order was chosen such as to be able to collectpossible higher-order nonlinearities in the model. Al-though lower-order approximations might also proveto be sufficient, such an analysis was out of the scopeof the present work.For the dynamic gas model, we regard the ideal gaslaw and the polytropic equation pV   =  mRT,  (3) pρ − γ  =  const. ,  (4)where  ρ ,  R ,  T  ,  V   and  m  denote the density, gas con-stant, temperature, volume of the fluidic muscle andgas mass, respectively, and  γ   is the polytropic ex-ponent, taken here as  γ   =  1 . 4. After taking the timederivative of (4) and back-substituting (4), one obtains ˙ p  = γ RT V  ˙ m − γ pV  ˙ V.  (5)Note that in this derivation all thermodynamical vari-ables, including temperature, are assumed to changewith time.For the performed actuations it can be assumedthat the volume of the fluidic muscle only depends onthe stroke. We approximate this volume-stroke depen-dency displayed in Fig. 8 by a third order polynomial V(s) = 3  i = 0 b i s i (6)leading to the time derivative ˙ V   =  d V(s) d t  =  d V(s) d s ˙ s  =   3  i = 1 ib i s i − 1  ˙ s.  (7)The mass flow  ˙ m  with which the muscle is inflatedhas a nonlinear characteristic and is a function of pres- Fig. 8  Volume of pneumatic muscle over stroke sure  p  inside the muscle and the input voltage  u  of thevalve ˙ m = φ(p,u),  (8)where the supply pressure is assumed to be constantand the function  φ(p,u)  is still unknown.By substituting (8) and (7) into (5), one obtains the representation for  ˙ p ˙ p  = γ RT V(s)φ(p,u)     f  1 (s,p,u) − γ pV(s) d V(s) d s ˙ s     f  2 (s,p, ˙ s) (9)which can be viewed as a superposition of two terms f  1 (s,p,u)  and  f  2 (s,p,  ˙ s) .In (9), the second term can be readily computedoncetheapproximation V(s) isknown(frommeasure-ments) and the pressure is given.The difficult term in (9) is the first term, as the non-linear valve function  φ(p,u)  is problematic to mea-sure. In the sequel, we propose an alternative way of determining this term, based on an interpretation thatmakes its measurement easier.Assume the position of the actuator to be constant.Then, for the velocity it holds  ˙ s  =  0, and the secondterm of the right side of the equation vanishes. Thusthe first term in (9) is just the pressure rate for fixed actuator length and given voltage and pressure values ˆ˙ p s (p,u) = ˙ p(s,p,u) | s = const . .  (10)In order to determine this two-parametric func-tion, the actuator is fixed in a rigid frame at differ-
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