IROS2019国际学术会议论文集 2592

IEEE TRANSACTIONS ON ROBOTICS VOL 34 NO 6 DECEMBER 20181643 Clock Torqued Rolling SLIP Model and Its Application to Variable Speed Running in a Hexapod Robot Wei Chun Lu Ming Yuan Yu and Pei Chun Lin Abstract In this paper we report on the development of the clock torqued rolling spring loaded inverted pendulum CTR SLIP model The new model which adds clock based torque control on the leg orientation of the previously developed R SLIP model has two advantages fi rst reg ulating the model to follow its passive dynamic running i e at a fi xed point signifi cantly increases the model s basin of attraction and second formulating the model closer to the empirical robot enables the model to serve as the transient and steady state running template of the robot as the anchor These features enable the model robot to perform speed transition from one fi xed point profi le to another and the experimental validation confi rms that the robot can successfully transition between two running speeds bidirectionally The achievement of variable speed running by the proposed method has a unique merit it is purely model based and there is no need for further tuning optimization or learning processes Regard ing the robot the proposed strategy only requires it to have simple position controltoregulateitslegorientations andthereisnoneedforothersensory modules to provide information for feedback IndexTerms Leggedrobot running variable speed clock model fi xed point basin of attraction BOA I INTRODUCTION Evolution has led to legged animals that have complex morphol ogy with high degrees of freedom DOFs and with adequate sens ing actuation and control they are capable of performing agile ro bust and dynamic locomotion on a wide range of terrains Because of their complexity in every aspect it is extremely diffi cult to ex plore the underlying mechanism of their locomotion in a precise and detailed manner However for certain types of motion such as run ning the researchers did fi nd that the legged animals seem to share a common motion pattern that can be represented by a simple spring loaded inverted pendulum SLIP 1 2 which is composed of a point mass and a massless linear spring Later Full and Koditschek proposed the template and anchor concept motion of the complex animals as the anchors can be extracted by reduced order dynamic models as the templates e g SLIP and on the other hand the latter ManuscriptreceivedMay15 2018 acceptedJuly1 2018 Dateofpublication October 4 2018 date of current version December 4 2018 This paper was recommended for publication by Associate Editor A Degani and Editor A Kheddar upon evaluation of the reviewers comments This work was supported by the Ministry of Science and Technology Taiwan under Contract MOST 107 2634 F 002 004 Corresponding author Pei Chun Lin W C Lu and P C Lin are with the Department of Mechanical Engineering National Taiwan University Taipei 10617 Taiwan e mail r05522807 ntu edu tw peichunlin ntu edu tw M Y Yu was with the National Taiwan University Taipei 10617 Taiwan He is now with the Robotics Program University of Michigan Ann Arbor MI 48105 USA e mail myyu umich edu This paper has supplementary downloadable multimedia material available at http ieeexplore ieee org provided by the authors This includes videos rep resenting the simulation and experimental results from within the article This material is 39 4 MB in size Color versions of one or more of the fi gures in this paper are available online at http ieeexplore ieee org Digital Object Identifi er 10 1109 TRO 2018 2862903 serves as the control guidance to excite the dynamic locomotion of the former 3 Various reduced order models have been reported in the last two decades for example a SLIP R model that adds a rolling foot to the SLIP model 4 a clock torqued SLIP CT SLIP model where the leg is regulated by the clocked torque 5 a TD SLIP HIP SLIP model that adds hip torque and a leg damper to the SLIP model 6 7 a TD SLIP model on compliant terrain 8 an active SLIP model 9 and a two segmented leg model 10 In addition some models have been utilized to control the robots for example the SLIP controller embedded in a planar hexapod robot model for pronking 11 an axial torsional SLIP model in a bipedal model for transitioning between walking and running 12 a bipedal spring mass model to control a biped robot ATRIAS 13 Previously we had developed a reduced order rolling SLIP model R SLIP that served as a template for initiating stable running on a RHex style 14 robot acting as the anchor 15 By utilizing the fi xed point trajectories of the R SLIP model as the robot leg trajectory reference the robot can successfully initiate stable running at various speeds While the R SLIP model adequately serves as the stable run ning template for the robot it cannot capture the transient behavior of the robot owing to a fundamental difference between the model and the robot in energy setting i e an unactuated leg versus actuated legs Here we report on the development of the clock torqued rolling SLIP CTR SLIP model which adds clock based torque control on the leg orientation of the previously developed R SLIP model The CT SLIP model has been reported before 5 where the clocked run ning trajectory requires a user predefi ned 4 parameter leg trajectory i e Buehler Clock 14 obtained by optimization Here the model s passive dynamic motion i e fi xed point trajectory is directly utilized asthereferenceclockedtrajectory sothemodelitselffunctionsbothas trajectory generator and trajectory controller The added leg regulation of the CTR SLIP model not only greatly increases basin of attraction BOA of the fi xed point in comparison with that of the R SLIP model but also provides better mapping between the template and the anchor so the former can represent both the transient and steady state behavior ofthelatter Thisaddedfeatureallowsustoexplorevariable speedrun ningbyswitchingamongfi xedpointswithdifferentdesignatedspeeds Variable speedrunninghasbeenreportedinasingleleggedplanarhop per by controlling both DOFs 16 or by modulating the applied torque 17 However tothebestofourknowledge therearenoreportsregard ing variable speed running in untethered multilegged robots guided by the extremely simple reduced order model with a clocked open loop strategy The achievement of variable speed running in the model and robot by the proposed method has a unique merit it is purely model based and there is no need for further tuning optimizing or learning processes With regards to the robot the proposed strategy only requires the robot to have simple position control to regulate its leg orientations no need for other sensory modules to provide infor mation for feedback 1552 3098 2018 IEEE Personal use is permitted but republication redistribution requires IEEE permission See http www ieee org publications standards publications rights index html for more information IEEE Transactions on Robotics T RO paper presented at IROS 2019 It should be cited as a T RO paper 1644IEEE TRANSACTIONS ON ROBOTICS VOL 34 NO 6 DECEMBER 2018 Fig 1 a R SLIP model and b the CTR SLIP model The symbols m l r kt v and represent intrinsic parameters touchdown states and generalized coordinates respectively The intrinsic parameters are set at m 7 78kg r 0 075m l 0 082m kt 22 8N m rad in simulation matching the physical specifi cations of the robot c utilized in experimental validation Theremainderofthepaperisorganizedasfollows SectionIIreports on the development of the CTR SLIP model Section III describes the simulation results of the model Section IV reports on the experimental results and Section V concludes the work II DEVELOPMENT OF THECTR SLIP MODEL A Review of the R SLIP Model Similar to the well known SLIP model the R SLIP model as shown inFig 1 a isareduced order conservative sagittal plane one legged model that is capable of performing stable running motion i e with alternating stance and fl ight phases 15 The R SLIP model has a torsionspringlegwitharollingcontactonthegroundwithoutslippage so the equivalent linear stiffness varies while the model leg rolls The R SLIP model has four intrinsic parameters including mass m bar length l radius of the circular rim r and torsion spring stiffness kt as shown in Fig 1 a The equations of motion EOM of the R SLIP model in the stance phase can be derived by using the Lagrangian method d dt L qi L qi qi qi 1 where the symbols L and represent Lagrangian energy supplied into the system and the Rayleigh dissipation function respectively For the R SLIP model the latter two terms are both zeros owing to its conservative nature The model can be parameterized by two general ized coordinates q T as shown in Fig 1 a representing the leg and the torsion spring confi gurations Thus the Lagrangian of the model L can further be expressed as L q q T V 1 2 m x2 z2 1 2kt 0 2 mgz 2 where the fore aft and vertical displacements of the point mass x z are functions of the generalized coordinates x r 0 0 rcos lcos z r rsin lsin 3 and the subscript 0 indicates the initial natural condition The motion of the R SLIP in its stance phase can be numerically generated when the touchdown states of the model are provided including touchdown speed v touchdown angle and landing angle as shown in Fig 1 a AftertheR SLIPmodeltakesoff themodelentersitsballistic fl ight phase and the leg is reposed to the designed landing angle The stability property of the model can be analyzed by using the Poincar e return map The touchdown event is chosen as the Poincar e section and three touchdown states v are considered Because Fig 2 Fixed points of the R SLIP model The circular and cross marks rep resent stable and unstable fi xed points respectively a Distribution of the fi xed pointsoftheR SLIPmodelwithvarioustouchdownspeeds v b Basinsofat traction shadedregion ofthemodelwithtouchdownspeedsv 1 5m s blue and v 2m s green at touchdown is designed to fi x at a specifi c value and the model is conservative v is the same for all steps Therefore only is utilized for return map analysis and it is qualifi ed as a fi xed point if n 1 n 4 In this 1 D case eigenvalues of the corresponding Jacobian matrix are equal to the slope conditions of the fi xed points If absolute value of the slope at the fi xed point is smaller than one the fi xed point is stable Fig 2 a shows the distribution of the stable and unstable fi xed points of the R SLIP model with various touchdown speeds as the exemplary demonstration When v 1 1 25m s all the fi xed points are unstable As the speed increases stable fi xed points begin to appear intheregionswithsmall 5 25 wherethemodelexhibitsgeneral forward running behavior The details of the R SLIP behaviors can be found in 15 B CTR SLIP Model The fi xed point trajectory of the R SLIP model represents the ideal passivedynamics ofthe system However unlikethemodel s effortless repositioning of the massless leg the empirical robot usually needs actuators to generate the desired leg motion profi le The proposed CTR SLIP model aims to include this realistic setting such that the model i e thetemplate andtherobot i e theanchor canbemapped better Thus the dynamic behavior analysis can be extended from the steady state motion as reported in 15 to the transient behavior More specifi cally the added clock torque of the CTR SLIP model tries to regulate the model s leg motion t i e one of the generalized coordinate of the model to follow the fi xed point trajectory of the R SLIPmodel fi xedpoint t asshowninFig 1 b Inthiscase themodel thatstartswithothertouchdownstatesorisdisturbedtootherstatescan beregulatedbacktothedesired passivedynamics ofthemodelitself Moreover because of this added control feature the transient behavior of the model can be utilized as the guideline for understanding the IEEE Transactions on Robotics T RO paper presented at IROS 2019 It should be cited as a T RO paper IEEE TRANSACTIONS ON ROBOTICS VOL 34 NO 6 DECEMBER 20181645 empirical robot s motion thus achieving variable speed running of the robot in real time The CTR SLIP model utilizes position control to modulate the leg motion because of thefollowing two reasons 1 Thefi xed point trajec tory of the R SLIP model has a specifi c leg orientation profi le versus time fi xedpoint t so it is reasonable to use position control to regulate the leg motion 2 The model has a compliant leg that can adapt to environment variation so it does not have high demand of the force control as other models with rigid legs do The actuator motor is a torque source so the position control of the model is set as kP f ixed point kD f ixed point 5 wherekPandkDaretheproportionalandderivativegains respectively ThesimplePDcontrolisutilizedbecauseitisthemethodthattherobot uses Becausethemodelonlyhasapointmass i e amasslessleg PD control with a wide selection of gains can already track the desired leg profi le well Theaddedclock torque oftheCTR SLIPmodelregulates one ofthe generalized coordinates i e the leg motion while the other i e the spring confi guration is passively determined by the model dynamics Thus the EOM of the model in stance phase can be represented as d dt L L d dt L L 0 6 The expended EOMs are ml2 2mrl cos sin 2mr2 1 sin mrl cos sin 2mr2 1 sin mr2 2cos mrl 2 2 sin mrl 2cos mg lcos rcos kP f ixed point kD f ixed point 7 and mrl cos sin 2mr2 1 sin 2mr2 1 sin mr2 2cos mrl 2 cos sin mgrcos kt 0 0 8 The model is no longer conservative due to the presence of the PD controller so the dynamics of the CTR SLIP model in its stance phase are expected to be different from that of the R SLIP model In contrast the dynamics of the model in fl ight phase are the same as the latter the ballistic fl ight Here we are interested in utilizing the CTR SLIP model as the template to initiate general running of the multilegged robots so the fi nal model is composed of two CTR SLIP legs whose motion profi les are shifts with one gait stride period as shown in Fig 3 a Hereafter it is referred to as the two leg CTR SLIP model The two virtual legs contact the ground alternately and periodically this generalized settingallowsthemodeltobedeployedtorobotswithdifferentnumbers of legs to facilitate bipedal running quadrupedal trotting and hexapod tripodrunning forexample Inaddition duetothealternatetouchdown of the legs each leg s aerial phase duration is equal to two fl ight phase time plus one stance time of the model The extended aerial phase time duration of the leg makes the empirical trajectory implementation of the robot more feasible Fig 3 Two leg CTR SLIP model with fi xed point 1 5 m s 10 a Motion profi les of the legs versus time motorand motor b Distances of the lowest points of the legs relative to the mass point The boundaries of the relative phase ofeachlegcanbedeterminedbytheintersectionsofthehcurves c Illustration of the relative phase C Relative Phase The leg motion of the CTR SLIP model is designed to follow a realistic trapezoid speed profi le which is achievable by the robot as shown in Fig 3 a If the model runs at the fi xed point the leg touches down at the exact predicted time and with the exact predicted landing angle In contrast if the model does not run at the fi xed point i e in transient motion both the touchdown timing and landing angle are different to those of the model in steady fi xed point motion Thus a new parameter relative phase which has not been explored before plays an important role and needs to be introduced relativephase tTD T 9 where tTDand T represent the time offset of the touchdown timing andperiodofthefi xed pointtrajectory respectively Therelativephase shown in 9 is utilized to quantitatively identify the level of offset in a dimensionless manner As shown in Fig 3 c if the model operates at the fi xed point trajectory where it touches down exactly when it should be the relative phase is zero On the other hand when the model operates at other conditions the model may touchdown earlier or later than its designated timing and the relative phase is defi ned to be negative and positive respectively The relative phase of the CTR SLIP model has a bounded range which can be determined by confi gurations of the two legs relative to each other and relative to the ground The leg touchdown happens if both legs are in the air and the lowest point of this leg is lower than that of the other leg Fig 3 b shows the lowest points of the legs relative to the mass point h in one stride period The relative phase of each leg exists when the lowest point of this leg is lower than the other and it is 0 047 0 953 in the particular parameter set of the model robot Note that this range is the possible touchdown range of the leg not the range for the model to converge IEEE Transactions on Robotics T RO paper presented at IROS 2019 It should be cited as a T RO paper 1646IEEE TRANSACTIONS ON ROBOTICS VOL 34 NO 6 DECEMBER 2018 Fig 4 a c Transient behaviors of the R SLIP model and d f the CTR SLIP model a and d show the planar trajectories of the point mass of the models The numbers represent the motion of the models at their nth steps n 0 1 2 10 The red solid curves and blue dotted curves indicate the model in stance and fl ight respectively b and e show the touchdown states of the models at their nth steps The black dashed lines indicate the touch down states of the selected fi xed point c and f show the torque and power of the models for the fi rst fi ve steps III SIMULATIONMETHODS ANDRESULTS The characteristics of the CTR SLIP model were simulated and an alyzed by using commercial software Matlab R2016b MathWorks Inc Natick MA USA Two kinds of simulations were executed A multistep simulation for understanding the model s transient prop erty during convergence and a BOA simulation for overall fi xed point convergence characteristics A Multistep Simulation In this set of demonstrations the transient behavior of the R SLIP model i e passive dynamics and the CTR SLIP model i e with clocked torque ar