Parallel Robots: Theory and Applications Patrick Grosch Federico Thomas Parallel Robots With Unconventional Joints Kinematics and Motion Planning Parallel Robots: Theory and Applications Series Editors J. P. Merlet, Sophia Antipolis, France Sébastien Briot, Institut de Recherche en Communications et Cybernétique de Nantes, CNRS, Nantes, France P. Martinet, Nantes, France Since a few decades, parallel robots have attracted the attention of industrials and researchers. For industrials, they are interesting due to their inherent advantages which are their large payload-to-mass ratio, intrinsic stiffness properties and great acceleration capacities. For researchers, their inherent complexity leading to issues intermsofdesign,modellingandcontrolallowedthedefinitionofmanyinteresting scientific problems to be solved. We reached now a point where the research on parallel robots has ledto thedefinitionof a quantity ofscientificresultswhich, for the moment, miss to be compiled to be easily usable by researchers, teachers, professionals and students for their formation and future works. This collection aims at filling this gap, by proposing a series of book devoted to all the scientific and technological fields required for having a better understanding of the behavior of parallel robots and/or for helping the future parallel robot designer in his work. Therefore, we intend to cover all topics on parallel robots, from their (geometric, kinematic, dynamic, elastic, etc.) modelling, to their advanced control, by also considering the singularity analysis, the calibration and model parameter identi- fication problems, the definition of advanced algebraic tools necessary for their study, their optimal design, and many other scientific and technological aspects. Springer and the series editors welcome book proposals. Potential authors are invited to contact Nathalie Jacobs, Executive Publishing Editor Engineering, at [email protected] More information about this series at http://www.springer.com/series/13855 Patrick Grosch Federico Thomas (cid:129) Parallel Robots With Unconventional Joints Kinematics and Motion Planning 123 Patrick Grosch Federico Thomas Institut deRobòticai Informàtica Institut deRobòticai Informàtica Industrial, CSIC-UPC Industrial, CSIC-UPC Barcelona,Spain Barcelona,Spain ISSN 2524-6232 ISSN 2524-6240 (electronic) Parallel Robots: Theory andApplications ISBN978-3-030-11303-2 ISBN978-3-030-11304-9 (eBook) https://doi.org/10.1007/978-3-030-11304-9 LibraryofCongressControlNumber:2019932702 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Inthisbook,itisshownhowthesubstitutionofordinaryjointswithunconventional joints, such as lockable revolute or non-holonomic spherical joints, in lower mobilityparallelrobotsprovidethemwithfascinatingfeatures.Themostimportant of them being the possibility of approximating trajectories for their moving plat- form in a configuration space of higher dimension than the number of their con- tinuously actuated joints. More specifically, it is shown how the introduction of these unconventional joints allows a parallel robot with an instantaneous kine- maticsequivalenttothatofalowermobilityrobottoattainanyconfigurationwithin asix-dimensionalworkspace,thatis,tobehavesimilarlytoafull-mobilityrobotat the cost of introducing maneuvers. After presentingageneral kinematicframeworkfor lockablerevolutejoints (R b joints, for short) and non-holonomic spherical joints (S joints, for short), three n particular parallel robots are analyzed in detail; namely (cid:129) the 4R RPS spatial reconfigurable robot (Chap. 2), b (cid:129) the 3S PU spatial non-holonomic robot (Chaps. 4 and 5), and n (cid:129) the S-2SPS spherical non-holonomic robot (Chaps. 6 and 7). The study of each of these three robots has been conducted with the greatest possible generality. In general, the performed analyses include (cid:129) the computation of its direct and inverse kinematics (both in position and velocity), (cid:129) the analysis of its singularities, (cid:129) the computation of its workspace, (cid:129) the design and implementation of a motion planning algorithm, and (cid:129) the implementation of a prototype to validate the theoretical results. The methods used to perform the above analyses and computations, and the algorithms proposed to solve the path planning problems they originate, cover a widerangeofsituationsthatcanbeusedasareferencetootherparallelrobotswith the aforementioned unconventional joints. v vi Preface The presented path planners are open-loop methods based on first-order kine- matic models (the dynamics of the system is not considered). This obviously simplifies the problem to the point in which most of the presented mathematical models are derived from elementary geometrical and kinematical arguments. Nevertheless, these models are enough for many practical applications where inertial and gravity forces can be neglected compared to the forces exerted by the actuators, and provided that we are able to construct the input as a function of the system state to compensate for noises and errors in the system. Themethodsandresultspresentedinthisbookshouldbeofinteresttopracticing and research engineers as well as Ph.D. students from the field of mechanical engineering and, in particular, from robotics. All the material presented here is the result of a research started in 2008, when Patrick Grosch joined the Institut de Robòtica and Informàtica Industrial (CSIC-UPC)asdevelopmentengineer.DuetofinancialshortagesthatdrovePatrick to work in many different development projects, this research spanned over more than8years.After alotofsetbacks thatchallengedourcapabilities andpushedour patience to the limit, this research eventually led Patrick to achieve the Ph.D. in 2016. The encountered difficulties were alleviated thanks to the kind help of our colleagues. In particular, we want to mention Raffaele Di Gregorio, from the University of Ferrara, Italy, who invited us to spend a fruitful research stay in his laboratory. We also want to extend our gratitude to Krzysztof Tchoń and Janusz Jakubiak,fromtheUniversityofWrocław,Poland.Withouttheirhelpsomepartsof this book would not have been possible. Barcelona, Spain Federico Thomas May 2018 Contents 1 Introduction: Lockable and Non-holonomic Joints . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Reconfigurable Robots with Lockable Joints . . . . . . . . . . . 6 1.1.2 Underactuated Robots with Non-holonomic Joints. . . . . . . 6 1.2 Precursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Precursors of Robots with Lockable Joints . . . . . . . . . . . . 9 1.2.2 Precursors of Robots with Non-holonomic Joints. . . . . . . . 10 1.3 Organization of this Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Parallel Robots with Lockable Revolute Joints. . . . . . . . . . . . . . . . . 19 2.1 Kinematics of the 4R RPS Parallel Robot . . . . . . . . . . . . . . . . . . 21 b 2.1.1 Position Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.2 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Motion Planning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Generating the Road Map . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Finding a Path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Hardware Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Software Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Spherical Non-holonomic Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 Under-Actuated Parallel Robots with Spherical Non-holonomic Joints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Implementation of Spherical Non-holonomic Joints . . . . . . . . . . . 38 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4 Kinematics of the 3S PU Spatial Robot . . . . . . . . . . . . . . . . . . . . . . 43 n 4.1 The 3S PU Robot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 n 4.2 Instantaneous Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 vii viii Contents 4.3 Statics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4 Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 Motion Planning for the 3S PU Robot . . . . . . . . . . . . . . . . . . . . . . . 59 n 5.1 Motion Planning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2 Using Truncated Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 Kinematics of the S -2UPS Spherical Robot . . . . . . . . . . . . . . . . . . . 67 n 6.1 Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 Deriving a Bilinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.3 Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.4 A, B, and Rotations in R4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.5 Workspace Computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.5.1 Graphical Representation of the Platform Orientation. . . . . 79 6.5.2 Workspace Boundaries Due to Singularities . . . . . . . . . . . 80 6.5.3 Workspace Boundaries Due to Joint Limits. . . . . . . . . . . . 82 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7 Motion Planning for the S -2UPS Robot. . . . . . . . . . . . . . . . . . . . . . 87 n 7.1 Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.2 Three-Move Motion Planner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.3 Two-Move Motion Planner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.4 Single-Move Motion Planner. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.5 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.5.1 Three-Move Motion Planner. . . . . . . . . . . . . . . . . . . . . . . 96 7.5.2 Two-Move Motion Planner . . . . . . . . . . . . . . . . . . . . . . . 96 7.5.3 Single-Move Motion Planner . . . . . . . . . . . . . . . . . . . . . . 97 7.5.4 Comparing the Three Motion Planners . . . . . . . . . . . . . . . 99 7.6 Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter 1 Introduction: Lockable and Non-holonomic Joints 1.1 Motivation Aserialrobotisasetofrigidbodies,orlinks,connectedinseriesthroughactuated joints,whicharetypicallyeitherrevolute(R)orprismatic(P)joints.Oneendofthis serial chain of links is called the base and the other the end-effector. In a parallel robot,theend-effector(alsoknowninthiscaseasthemovingplatform)isconnected to the fixed base through several serial chains. In this case, most of the joints are notactuated.Thesepassivejointsaretypicallyeitheruniversal(U)orspherical(S) joints. Prismatic, revolute, universal, and spherical joints constitute the four major jointsusedinrobotics(Fig.1.1). TheoriginofparallelrobotsisattributedtoGoughandWhitehall[16](Fig.1.2) andStewart[37].TheirworkledtowhatisnowadaysknownastheGough–Stewart platform.In1965,Stewartformalizedtheconceptsthatdefineaparallelrobot.Since then,manydevelopmentsandstudieshavebeencarriedoutonthiskindofrobots. Innocenti and Parenti-Castelli [21] and Dasgupta and Mruthyunjaya [11] give an introductoryoverviewofthedifferentkindsofparallelrobots.Thereferencescom- piledin[9, 27]permittoestablishagoodstartingpointwheretofindmethodsfor thecomputationofthedirectandinversekinematics,theconfigurationspaceandthe singularities,themanipulabilityandaccuracy,etc.,ofdifferentparallelrobots. TheGough–Stewartplatformremainsasthemostpopularspatialparallelrobot. It is commonly seen in most motion simulators. This robot is said to have a 6SPS architecture,meaningthateachofits6legsconsistsofaprismaticjointconnected to the fixed base and to the moving platform through passive spherical joints. The underline is used to denote that the joint is actuated. This parallel robot is said to be a fully-parallel robot because: (a) it has as many serial chains (known as legs) asthenumberofdegreesoffreedomofitsmovingplatform,(b)eachlegpossesses onlyoneactuatedjoint,and(c)nolinkofthelegsislinkedtomorethantwobodies [28, 34]. ©SpringerNatureSwitzerlandAG2019 1 P.GroschandF.Thomas,ParallelRobotsWithUnconventionalJoints, ParallelRobots:TheoryandApplications, https://doi.org/10.1007/978-3-030-11304-9_1