Aarts, Ronald

Cornelissen, Robin

Müller, Andreas

English

Due to their deterministic behaviour, compliant mechanisms are well-suited for high-precision applications. In this paper the benefits of redundant links and actuation are investigated in terms of increasing support stiffness and homogenising actuator loads. The manipulator is modelled with lumped inertia properties of the links and non-linear relations for the joint stiffnesses. The lumped parameter model allows a fast system level performance optimisation of the joint geometry simultaneously exploiting joint pre-bending and preloading, where the stiffness matrices of all joints are computed numerically efficient with non-linear flexible beam elements. This model is applied to optimise the design of a compliant and redundantly actuated 2-DOF 3RRR parallel kinematic manipulator. The improvement of support stiffness is demonstrated with an analysis of the first parasitic natural frequency. Balancing of the actuator torques is concluded from a potential energy analysis.</p

University of Twente Research Information

A Compliant and Redundantly Actuated2-DOF 3RRR PKM: Bestof Both Worlds?Robin Cornelissen1, Andreas Müller2, and Ronald Aarts1(B)1 Faculty of Engineering Technology, Structural Dynamics, Acoustics and Control,University of Twente, P.O. Box 217, 7500 AE Enschede, The NetherlandsR.J.Cornelissen@student.utwente.nl, R.G.K.M.Aarts@utwente.nl2 Institute of Robotics, Johannes Kepler University Linz,Altenbergerstraße 69, 4040 Linz, Austriaa.mueller@jku.atAbstract. Due to their deterministic behaviour, compliant mechanismsare well-suited for high-precision applications. In this paper the benefitsof redundant links and actuation are investigated in terms of increasingsupport stiffness and homogenising actuator loads.The manipulator is modelled with lumped inertia properties of thelinks and non-linear relations for the joint stiffnesses. The lumped param-eter model allows a fast system level performance optimisation of thejoint geometry simultaneously exploiting joint pre-bending and preload-ing, where the stiffness matrices of all joints are computed numericallyefficient with non-linear flexible beam elements.This model is applied to optimise the design of a compliant andredundantly actuated 2-DOF 3RRR parallel kinematic manipulator. Theimprovement of support stiffness is demonstrated with an analysis of thefirst parasitic natural frequency. Balancing of the actuator torques isconcluded from a potential energy analysis.1 IntroductionIn precision applications compliant mechanisms, or more precisely flexure-basedmechanisms, are frequently used where the motion is enabled by elastic defor-mation of slender elements. Deterministic behaviour is ensured by the absence offriction, hysteresis and backlash. However, in spite of significant recent achieve-ments in terms of the range of motion [6], it remains a challenge to maintain ahigh support stiffness, i.e. stiffness in directions in which the mechanism is notsupposed to move, especially when the flexure joints in the mechanism undergolarge deflections.Redundantly actuated parallel kinematic manipulators (PKM) are researchedto combine the advantages of PKM, i.e. the high stiffness, low inertia and largeaccelerations, with an improved handling of singularities and optimised actuatorc© Springer Nature Switzerland AG 2020A. Kecskeméthy and F. Geu Flores (Eds.): ECCOMAS 2019, COMPUTMETHODS 53, pp. 163–171, 2020.https://doi.org/10.1007/978-3-030-23132-3_20164 R. Cornelissen et al.loading made possible by the redundancy [3,5]. In [4] the design of an experi-mental planar 2-DOF test setup with three actuators has been presented. Theplatform is well suited to evaluate models and control strategies.The main goal of the present paper is to investigate whether the combina-tion of compliant joints and redundant actuation can be exploited to furtherimprove the dynamic properties of PKM, while circumventing the drawbacks ofcompliant joints. More specifically, the support stiffness from additional linksis expected to reduce the decrease of this stiffness commonly observed at largejoint angles. Furthermore, the redundant actuation offers a possibility to combineload balancing techniques with preloading of the compliant joints to balance theactuator effort, which is needed to position the end-effector (EE) at any positionexcept for the equilibrium position, i.e. the centre of the workspace.For this analysis the manipulator has to be modelled taking the non-linearstiffness of the joints into account. Such joint models are usually rather com-plicated and hence computationally too expensive when used to optimise themanipulator dynamic performance. For this reason many studies, including [6,7],investigate a single compliant joint. We propose to describe the complete manip-ulator with lumped inertia properties of the links and non-linear relations forthe stiffness matrices of the joints. The lumped model allows a fast systemlevel performance evaluation that is used to determine optimal joint orienta-tion, pre-bending and preloading. The stiffness matrix of each joint depends onthe joint geometry. It is computed throughout the full operating range of thejoint where leaf springs are modelled with the non-linear flexible beam elementsof the SPACAR software package [2]. Next piecewise linear interpolation is usedto approximate all coefficients in this matrix as functions of the joint angle.The main part of this paper are the analysis and optimisation methods forthe support stiffness, Sect. 3, and actuator torques, Sect. 4. These are applied tothe example manipulater introduced in Sect. 2. The results are presented anddiscussed in Sect. 5 with conclusions in Sect. 6.2 Example ManipulatorThe dynamic and kinetostatic models developed in the next sections will bedemonstrated for a planar 2-DOF 3RRR PKM equipped with compliant joints,Fig. 1(a). The three arms of the manipulator are similar but rotated 120◦ relativeto each other. The shoulder joints and actuators are located at the verticesof an equilateral triangle, Fig. 1(b). This figure illustrates the basic geometricparameters like the total length L of each arm and the distance R of each actuatorto the centre of the triangle. The workspace reachable by the EE is bounded bythree circular arcs with radii L as indicated with the unshaded area in the figure.In order to reach all points within this workspace, the joints have to allow jointangles in some range. The ranges for the shoulder and elbow joints are indicatedin the figure with the angles θrs and θre, respectively. These angles depend onthe ratio L/R and the division of the total arm length L into the upper armA Compliant and Redundantly Actuated 2-DOF 3RRR PKM 165θL θll21eshrrφwsrR(a) Schematic manipulator overview. (b) Workspace.Fig. 1. Planar 2-DOF 3RRR PKM similar to [4] except for the use of compliant joints,Fig. 2.length l1 and forearm length l2. For the manipulator considered in this paperthese dimensions areR = 0.2305 m, L = 0.2536 m, l1 = l2 = L/2. (1)Then the joint angles ranges, with the lower and upper limit between brackets,are:θrs = 50◦(−17◦...33◦), θre = 83◦(−49◦...34◦), θrw = 50◦(−22◦...28◦). (2)All compliant joints used in this example manipulator are butterfly joints,Fig. 2 [1]. For the shoulder joints this joint type is selected as it is known for itssmall pivot shift which makes it relatively easy to connect a common rotational(a) Elbow joint (pre-bended). (b) Wrist joint.Fig. 2. Two of the three compliant butterfly joint of the planar 2-DOF 3RRR PKM.The “intermediate” body that rigidly connects two or three connections between leafsprings are positioned on the top and bottom of the hinge and the top part is trans-parent in the images.166 R. Cornelissen et al.actuator to the upper arm. The elbow joints exhibit rather large rotation anglesfor which other joint types may be more favourable [6], but also a butterfly jointwill be used to limit the overall complexity of the manipulator, Fig. 2(a). Finally,the wrist needs some special attention as it has to connect three links, Fig. 2(b).To maintain the symmetry as well as sufficient support stiffness, butterfly jointsconnect each forearm to an intermediate body. This extra body introduces anadditional degree of freedom with a vibration mode with a relatively low fre-quency that has to be taken into account in the dynamic analysis of Sect. 3.3 Support Stiffness HomogenisationThe support stiffness of compliant manipulators is often evaluated from thesystem’s natural frequencies as in this way an implicit scaling of translationaland rotational stifnesses is obtained. For a 2-DOF manipulator the first andsecond natural frequencies should be low as these are associated with the motionof the intended two degrees of freedom. The third natural frequency is the firstparasitic frequency that should be as high as possible throughout the entireworkspace. These natural frequencies can be computed with a detailed non-linear model of the full manipulator in which the EE position is varied in itsentire workspace. For a design optimisation the evaluation of many alternativedesigns will be needed and such model would be too computationally intensive.Hence, a simplified lumped model is proposed.This lumped parameter model should account for the mass properties of therigid links and the stiffnesses of the joints. For the example manipulator of Sect. 2seven rigid bodies can be identified, i.e. three upper arms, three forearms andthe intermediate body connecting all forearms. Each body is characterised byits 6 × 6 mass matrix that is constant in a local frame connected to the body.The links are connected to each other and the ground with nine joints in theshoulders, elbows and wrist. For each joint its full 6 × 6 stiffness matrix shouldbe considered. For a specific joint with known dimensions this matrix can beevaluated numerically e.g. in the joint’s pivot in the undeflected state. The non-linear flexible beam elements implemented in the SPACAR software package[2] have proven to be well-suited for efficient modelling of various compliantjoint types [6,7]. Even when large deflections are considered three or four beamelements suffice for each leaf spring. Some known limitations arising from thebeam model can be corrected like the torsional stiffening due to constrainedwarping in short and wide leaf springs [7]. With this model the stiffness matrixof each joint is computed throughout the full operating range of the joint andpiecewise linear interpolation is used to approximate its coefficients as functionsof the joint angle.At system level all degrees of freedom of the rigid links are combined. In theexample manipulator the seven links have 42 degrees of freedom in total. Fora straightforward eigenvalue analysis it is proposed to define these independentcoordinates in a coordinate frame located at the EE and aligned with the globalcoordinate frame. For the system level mass matrix the link mass matrices Muppi,A Compliant and Redundantly Actuated 2-DOF 3RRR PKM 167Mfori, Mint for upper arm i, forearm i and intermediate body respectively arecombined to a block diagonal matrix M. Similarly, the joint stiffness matrices arecombined by adding the stiffness matrices Ksi, Kei, Kwi of shoulder i, elbow iand wrist i respectively,K =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣Ks1 + Ke1 −Ke1 0 0 0 0 0−Ke1 Ke1 + Kw1 0 0 0 0 −Kw10 0 Ks2 + Ke2 −Ke2 0 0 00 0 −Ke2 Ke2 + Kw2 0 0 −Kw20 0 0 0 Ks3 + Ke3 −Ke3 00 0 0 0 −Ke3 Ke3 + Kw3 −Kw30 −Kw1 0 −Kw2 0 −Kw2 Kw123⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3)where Kw123 = Kw1 + Kw2 + Kw3. It should be noted that the link mass andjoint stiffness matrices are initially computed in a local frame attached to thelinks’ centre of mass or the joints’ pivots, respectively. For each EE position theinverse kinematic solution for all links and joints can be found relatively easy.With these geometric data all link and joint matrices can be transformed to theEE frame with pre and post multiplication with adjoint matrices that reflect theinvolved sequence of translations and rotations.Recall that for each joint the coefficients of the stiffness matrix are linearlyinterpolated as functions of the joint angle. In the transformation to the EEframe the joint orientations γs and γe of shoulder and elbow joints are designparameters. Additionally the shoulder and elbow joints can be manufacturedwith pre-bending angles βs and βe. These angles specify an offset between thejoint angles at which the EE is in its equilibrium position and at which the jointis in the undeflected configuration, i.e. exhibits the largest support stiffness. Inthe computation of the actual joint stiffness, these pre-bending angles have tobe added to the angles used in the linear interpolations.4 Actuator Torque Balancing and Potential EnergyFor the analysis of the support stiffness in the previous section, in particular thejoint stiffnesses in the stiff directions play a role. The compliant directions are lessrelevant as the lowest natural frequencies are discarded. However, the actuatortorques needed to position the EE depend on these joint stiffnesses and can becomputed from a kinetostatic analysis. Each joint is described by a stiffness ksi,kei or kwi in the driving direction, which is the lower right coefficient in therespective stiffness matrix of the previous section and is weakly dependent onthe joint angle. Furthermore, for the shoulder and wrist joints pretension anglesαs or αe are taken into account. This pretension angle is defined as the jointangle at which the joint is undeformed and hence no joint torque arises.For any EE position the motor torques τi (i = 1, 2, 3) acting on the shoul-der joints needed to keep the system statically balanced can be determinedfrom equilibrium equations that can be derived for all links. As the manipulatoris redundantly actuated, there is no unique solution for the actuator torques.168 R. Cornelissen et al.The null space solution is used to compute the actuator torques that minimisethe 2-norm of the vector τ with the three actuator torques τi [4]. It appears thatthis actuator norm ||τ || scales proportionally to the variation of the potentialenergy in the compliant joints throughout the workspace, which is relatively easyto evaluate in order to optimise the joint pretension angles αs and αe.5 ResultsThe analysis methods outlined in the previous sections will now be applied forthe design of the example manipulator proposed in Sect. 2. For high performancemetal links and joints would be preferred, but for a cost-effective first prototypeit is chosen to use 3D printed parts made by Selective Laser Sintering (SLS)of Nylone (PA2200). Typical material properties are its Young’s modulus E =1.7 GPa and density ρ = 930 kg/m3. Preloading of the joints is not possible withthis material as creep will result in rather quick unloading. Hence an alternativesolution for preloading should be applied like low-stiffness preloaded (metal)springs positioned parallel to the joints, as can be recognised in Fig. 1(a).Homogenised Support Stiffness. As outlined in Sect. 2 it was chosen touse butterfly hinges for all joints in the example manipulator. Butterfly hingeshave an extra internal degree of freedom due to rotation of an intermediatebody which may show a relatively low frequent vibration mode. This motioncan be constrained [1] when its frequency appears to be too low. As was alsopointed out in Sect. 2, the EE has an intermediate body as well for which thesame consideration applies. For the considered manipulator it appeared thatlightweight intermediate bodies can be used, such that the natural frequenciesof the internal modes were higher than some other parasitic mode and hence noprecautions had to be taken to constrain these modes.Figure 3 illustrates some steps in the optimisation procedure of which Table 1summarises some geometric parameters. In Fig. 3(a) the variation of the supportstiffness throughout the EE workspace is evaluated in terms of the first naturalfrequency for some initial design. For butterfly joints the reduction of the supportstiffness is relatively small as is shown in Fig. 4. Nevertheless the first parasiticnatural frequency varies more that a factor 2 throughout the workspace.Table 1. Geometric parameters of optimised joints. The vertical dimension of all hingesis 40 mm. The thickness of all leaf springs is 0.4 mm. The table shows the other in-planedimensions.Shoulder Elbow WristHeight[mm]Width[mm]Height[mm]Width[mm]Height[mm]Individual hinge optimisation 27.4 12.2 25.7 10.1 17.9Full optimisation 20.2 19.8 29.4 17.8 18.2A Compliant and Redundantly Actuated 2-DOF 3RRR PKM 169Fig. 3. Optimising the support stiffness, evaluated with the first parasitic natural fre-quency.With an optimal pre-bending and joint orientation this variation can bereduced as shown in Fig. 3(b), although the primary optimisation goal is increaseof the lowest frequency near the corners. Probably the redundant arm also offersadditional homogenisation of the stiffness.A further improvement is possible by optimising the joint parameters,Fig. 3(c), and finally combining all optimisation efforts, Fig. 3(d). In the finalresult, not only the lowest parasitic natural frequency has been increased, butalso its largest value near equilibrium is large again although this was not anoptimisation goal.Balanced Actuator Torques. In the design so far the actuator torques arenot considered. To illustrate the actuator torque balancing, Fig. 5 shows thepotential energy E stored in the compliant joints relative to the equilibriumconfiguration. The figure shows the steep increase in energy storage towards theextreme edges of the workspace without pretensioned compliant joints (αs =αe = 0◦). With pretension a significant reduction of the potential energy ispossible as is demonstrated in the figure with αs = 42◦, αe = −50◦. The torque170 R. Cornelissen et al.K55K44K66Fig. 4. Joint stiffness of the but-terfly shoulder hinge in driving(K66) and two support (K44, K55)directions.Fig. 5. Balanced potential energystorage illustrating the balancing ofthe driving torque.norm ||τ || scales similarly, hence a reduction of the driving torques with morethan an order of magnitude is achieved by this balancing.6 ConclusionIn this paper we propose an analysis method that enables a system level optimisa-tion of a parallel kinematic manipulator with redundant actuation and compliant(or flexure) joints. The goal of the optimisation is to combine the redundancywith preloading and pre-bending of the joints to obtain a high support stiffnessthroughout the workspace and to reduce the actuator torques. The approximatemodel used in this optimisation combines lumped mass properties of the rigidlinks with non-linear stiffness matrices of the joints computed as functions of thejoint angles.The method is applied to the design of a planar 2-DOF 3RRR parallel manip-ulator. It appears that orientation and pre-bending of the compliant joints canbe optimised to avoid a significant decrease of the support stiffness and preload-ing helps to lower the required actuator torques. The proposed design has beenmanufactured with 3D printing to experimentally verify the expected dynamicbehaviour, as will be published in the future.References1. Henein, S., Spanoudakis, P., Droz, S., Myklebust, L.I., Onillo, E.: Flexure pivot foraerospace mechanisms. In: Proceedings of the 10th European Space Mechanismsand Tribology Symposium, San Sebastian, Spain (2003)2. Jonker, J.B., Meijaard, J.P.: SPACAR - computer program for dynamic analysis offlexible spatial mechanisms and manipulators. In: Schiehlen, W. (ed.) MultibodySystems Handbook, pp. 123–143. Springer, Heidelberg (1990)A Compliant and Redundantly Actuated 2-DOF 3RRR PKM 1713. Kock, S., Schumacher, W.: A parallel x-y manipulator with actuation redundancyfor high-speed and active-stiffness applications. In: Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation, Leuven Belgium, pp. 2295–2300 (1998)4. Krajoski, K., Müller, A., Gattringer, H., Jörgl, M.: Design, modeling and controlof an experimental redundantly actuated parallel platform. In: Niel, K., Roth, P.M.(eds.) Proceedings of the OAGM&ARW Joint Workshop 2016 “Computer Visionand Robotics”, University of Applied Sciences Upper Austria, Wels Campus, pp.209–216 (2016)5. Müller, A., Hufnagel, T.: Model-based control of redundantly actuated parallelmanipulators in redundant coordinates. Robot. Auton. Syst. 60(4), 563–571 (2012)6. Naves, M., Brouwer, D.M., Aarts, R.G.K.M.: Building block-based spatial topol-ogy synthesis method for large-stroke flexure hinges. J. Mech. Robot. 9(4), 041006(2017). 9 pages7. Wiersma, D.H., Boer, S.E., Aarts, R.G.K.M., Brouwer, D.M.: Design and perfor-mance optimization of large stroke spatial flexures. J. Comput. Nonlinear Dyn. 9(1),011016 (2013). 10 pages

A Compliant and Redundantly Actuated 2-DOF 3RRR PKM:Best of Both Worlds?

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A Compliant and Redundantly Actuated 2-DOF 3RRR PKM:Best of Both Worlds?

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