MotionAnalysis of aParallel MobileRobot ShragaS ho val 1 and Moshe Sho ham 2 1 Department of IndustrialEng inee ring&M anage ment, AcademicCollege of Judeaand Samaria, Ariel,Israel shraga@yosh.ac.il 2 Faculty of MechanicalEngineering, Technion,Haifa, Israel, shoham@tx.technion.ac.il Abstract. This paper presents akinematicand forceanalysis of amobile robot built on the principle of parallel mechanisms.The robot consists of an upper plateconnected to 3 legs, eachequipped withanasynchronous driving unit.Akinematicmodel for the robot providesdatafor accurateposition estimate,even in rough and slippery terrains where conventionalodometryfails.The paper presents ananalysis of the forces actingon the robot under various surfaceconditions and robot configurations.This analysis provides usefuldatatodetermine whether a specificmotioncanbecompleted given the limitations on stability, the geometryand friction of the surface,and the required motion direction. The paper presents simulation results that are verified byexperiments using our prototype model. 1Introductio n Parallelmechanisms consist of anupperplatform that ismaneuvered by several (3-6)legs connected toalower stationary platform. The maneuverability, rigidity and accuracy arefunctions of the number of legs and the type of joints between the plates and the legs.The basicconceptualmechanics is knownas the Stewart platfo rm[Stewart,19 65]even tho ugh earlier versions arek nown,and sin ce the n many mani pulators wered evel ope dbased on t hi s me chani sm[Hunt19 83,T sai and Tahmasebi1983 and others]. Ben Horin and Shoham[1997]have suggested using mobilejoints between the legs and the stationary platform, turning the mechanismintoa semi-mobile robot.The mechanismconsists of the followingcomponents: three links of fixed length,havinga sphericaljoint on one endand a revolutejoint on the other end, three actuators whichmoveplenary on a stationary platformand anoutput platformhaving sixdegrees-of-freedom (DOF). Tofurther increasemobility,Ben Horin and Shoham[1999,2000] suggest amoreflexible design. This mechanism, showninFig. 1,is based on 3 inflatable legs,an upper platformand 3 asynchronous driving units for the three legs.The upper joint of eachleg is a revol ut ejoi nt, w hi le t he lo wer jo in t, whi chcon ne cts the leg t o t he driving uni t,is a sphericaljoint.This configuration offers six DOF for the upper plate where the control parameters are the positions (X,Y,Z)ofthe three driving units.The mechanismis designed for applications that requirealightweight and easy S. Yuta et al. (Eds.): Field and Service Robotics, STAR 24, pp. 323–331, 2006. © Springer-Verlag Berlin Heidelberg 2006 324S.Shovaland M.Shoham depl oyable robot.Given t he r eq ui red tr aje ctory fo r the en d ef fecter (6 parame ters) the inversekinematics model cangenerate the required pathofeachdriving unit, subject toits non-holonomicconstraints. Fig.1. The inflatable mobile robot [Ben Horin and Shoham, 2000] In this paper wepresent akinematicand forceanalysis for the parallel mo bile robot.The ki nem atica nalys is, s howni n s ectio n 2 ,en able s a ni mp roved position estimateba sed on addi tion ale ncode rs attache d to t he upp er pl atform. Section 3 details the forceanalysis for the robot,and section4 shows simulation results for motions in various configurations and con straints.Section5 verifies the simulationresults withexperiments conducted withour prototype platform and section 6 providesconcluding remarks. 2KinematicA nalysis Figure 2 is a schematicdescriptionof the parallel mobile mechanism. The upper pl ateis con ne cted toe acho f t he three leg s with r evol ut ej oi nts.The three le gs are driven by three asynchronous units thatareconnected to the legswith spherical joints.Controlledmotionof the three driving units determines the pose(position and orientation)of the upper plate. Motion Analys is of aParallel Mobile Robot 3 25 Fig.2. Sche maticdescription of the mobile parallel mechanism Todetermine the accurateconfiguration of the upper plate,both the absolute position of eachd rivin g unit (X i ,Y i ,Z i )and the directkinematics model are required .The position of t he driving uni ts is de termi ne dby Odo me tr yu sing encode rs attached t o t he drivin g w heels.Ta hmasebiand Tsai[19 94 ] s how that fo r the abovep arallel me chani smthe reare1 6 po ssible di rect ki ne matics sol ut ion s, which requireextensivecomputationaleffort.Furthermore,determining the absolutepositionofeachdriving unit is subjected toodometricerrors and cannot provide a reliable position estimate. Toimprove the accuracy and to simplify the directkinematics model Shovaland Shoham[2001] useadditionalmeasurements taken from on boardencoders attached to the upper revolutejoints.The additional encod ers me asure t he rotation ang le betwee n t he upp er pl ateand the legs.Based on the semeasu rem en ts, t he position s of the three driving u ni ts arederived in the U-V-Wcoordinate system (attached to the center of the upper plate). Ba sed on the positionof the drive units as determinedin the U-V-W system, the Euclidean distances between the drive units l 1 , l 2 and l 3 aregiven byEq. 1: 2 11 2 11 2 11 )()()( +−+−+− −+−+−= iiiiiii WWVVUUl (1 ) i –incyclicorder TheseEuclideandistances canalsobederived according to the position of the driving units as determinedby the Odometricmodel according toEq. 2. 2 11 2 11 2 11 )()()( +−+−+− −+−+−= iiiiiii ZZYYXXm (2) If the odometric system is accurate, the distances derived in the upper plate coordinate system ( l i )areequal t o t he di st ances derived in t he w orld coo rdi nate system ( m i ). If,however, thesedistances aredifferent, the odometriccalculation is faulty.Anodometricerror in one drive unit affects twodistances according to Eq. 2.Assuming a single odometricerror at eachinterval time, the accurate Driving Units +Spherical Joints Revolute Joi nts + Encode rs (X 1 ,Y 1 ,Z 1 ) (X 3 ,Y 3 ,Z 3 ) (X 2 ,Y 2 ,Z 2 ) 326 S.Shovaland M.Shoham position of three driving units is continuously updated.In the unlikely event that two(or three) driving units are subjected toodometricerrors simultaneously, the proposed procedurecannot beimplemented and additionalmeasures must be taken(i.e. re-calibrationof the robot’s position). 3 ForceAnalyses Fig. 3 shows the contact forces appliedon the robot by the ground.Doand Yang [1988] suggest a solution based on the Newton-Euler me thod, resultingin 36 linear equations.Other researchers use the principle of virtual work to reduce the complexity of the solution.Ben Horin [1999] uses the Kane method todetermine the dynamicequations, solvingit withnumericalprocedures.To simplify the solution weassume that the robot’s mass is concentrated at the upper plate, while the legs havenegligible mass.Wealsoassume that internalchanges in the robot internalconfiguration arequasistatic(dynamicforces associated withchanges of internalconfiguration arenegligible compared withother). a F 1 t F 1 a F 2 Y t F 2 a F 3 t F 3 X Z Fig.3. Contact forces applied on the robot The revolutejoints between the legs and the upper plate sustain only axial force-F a actin galong t he leg,and tangentialf orce– F t actin gp arallel t o t he axis of the revolutejoint.Reducing the problem to 6 unknownforces simplifies the solution to the following equations: x i ii t i i ii a ix maFFF =+= ∑∑∑ == 3 1 3 1 coscoscoscos γδµθ (3) y i ii t i i ii a iy maFFF =+= ∑∑∑ == 3 1 3 1 sincossincos γδµθ (4) ∑∑∑ == =+= 3 1 3 1 sinsin i zi t i i i a iz maFFF δθ (5) •• = =−+−−+= ∑∑ θδθγδµθ jrrFFrrFFM y cb ii t i a iz cb i i ii t iii a ix }))(sinsin()()sincossincos{( 3 1 (6) Motion Analys is of aParallel Mobile Robot 3 27 •• = =−++−+−= ∑∑ ξδθγδµθ jrrFFrrFFM x cb ii t i a iz cb i i ii t iii a iy }))(sinsin()()coscoscoscos{( 3 1 (7) •• = =−++−+−= ∑∑ ωγδµθγδµθ jrrFFrrFFM y cb iii t iii a ix cb i i ii t iii a iz }))(coscoscoscos()()sincossincos{( 3 1 (8) where a i F - the axialforceinleg I , t i F - the tangentialforceinleg i . θ i - the angle between leg i and the X-Yplane (thereforethe anglebetween the axialforceand the X-Yplane). µ i - the angle between leg i and the positiveXaxes. δ i - the angle between the tangentialforceofleg i and the X-Y. γ i -the angle between the tangentialforces of leg i and the positiveXaxes. b i r - the position vectorof the bottom of leg i . c r - the position vectorof center of the upper plate. a x ,a y ,a z –accelerations along the X,Y and Z θ , ω , ξ -Euler orien tatio nang le s J -Mom en t of in ert iaof t he upp er pl ate m –Mass of the upper plate Let us assume afriction coefficient µ between the driving wheels and the surface. We transformthe axial- a i F and tangential- t i F forces (derived by Eq. 3-8) toa new coordinate set definedby: d u ) -in t he di rectio no f t he req uired mo tio n, l u ) -in the lateraldirection,and n u ) -normal to the surface. The lateralforce F l actingon the driving unit is given by: ) ( l t il a il uFuFF )) •+•= (9) Toprevent lateral slippage, F l is bounded by: )( n t in a il uFuFF )) •+•< µ (10) The twoexpressions associated with F l provide the first conditionfor preventinglateral slippage: µ < •+• •+• )( )( l t il a i n t in a i uFuF uFuF )) )) (11) Given a specificmoment M d generated by the drivingmotorand r - the radius of the driving wheel, the frictiondrivingforce–F d generating the motionof the driving unit in the required directionis limited by: )( n t in a idd uFuFrMF )) •+•<= µ (12) Given a specific terrain topography and friction, the actual driving forces applied on eac hleg ca nbed etermi ne dbothalon g t he lon gi tu di naland laterald irectio ns. Furthermore,based on theseforces the accurate dynamic reaction of the robot canbecalculated. 328S.Shovaland M.Shoham 4SimulationResults Wefirst examine the forces applied on the robot when travelingat aconstant speedonahorizontal surface, withanidenticalinclination angles ( η )between the legs and the upper plate. The robot is travelingin the positiveYdirection witha configuration is showninFig.5a.We start the experiment with η =90 o (legsare perpendicular to the surfaceand the upper plate), while gradually and simultaneously reducing the inclinationangles for all legs.The weight of the upper plateis 100N.As expected, the axialforces - a i F on all three legs are identical, starting with33.33Nwhen the legs areperpendicular ( η =90 o ),and increasing as η decreases(Figure5b). Sinceall driving units arein the travel direction, the tangentialforces - t i F areclose tozeroinall legs,and thereforeare notshownin the graphs. 0 100 200 300 400 500 600 020406080100 Inclenatio n Angle-n [deg] Force [N] (a)( b) Fig.5. Axialforces for constant motionoverahorizontal surface. Next, we transform the axialand tangentialforces to the corresponding components in the longitudinal(F d ),lateral(F l ),and normal t o the surface di rection s ( F n ). Due to the symmetry, the normalcomponents in all legs remain con stant (33 .3N). The friction coe fficien t is 0.7,ge ne ratin gamaxim alf riction fo rceof23 .33Nfo r all le gs.Fig . 6 a s ho ws the lateraland lo ng itu di nalf orcesfo r legs 1and 2.Theseforces areidenticaldue to the symmetry of the twolegs relative to the drivingdirection. Fig.6b shows the same forces for leg 3.As shown, the lateralforces on legs1and 2 pass the maximalfrictionforceat η =53 o . The longitudinalforceofleg 3passes the frictionforcelimit at η =55 o .The robot can therefore travel at aconstant speedonahorizontal surface withafriction coefficient of 0.7 as long as the inclinationangleof the legs is larger than55 o .At that angle,longitudinal slippage occurs at leg 3.Further decreaseof the inclinationangle to53 o causes additionallateral slippage in legs 1and 2. Travel Direction 3 2 1 X Y Motion Analys is of aParallel Mobile Robot 3 29 Le g 1 an d le g 2 0 5 10 15 20 25 30 020406080100 n [deg] Force [N ] Fd Fl Fu Leg 3 -10 0 10 20 30 40 020406080100 n [deg] Fo rc e [N ] Fd Fl Fu (a)(b) Fig.6. Forces on legs 1and 2 (a)and 3 (b)for constant speed. In the next set of simulations, the robot travels along ahorizontal surfacein a straight line withconstant accelerationalong the Y+axis (equivalent to travelingataconstant speedonaninclined surface). Inaddition tofriction constraints,external stability must alsobeconsidered.Fig. 7 shows the forces on legs 1and 2 as afunction of the inclination angle η duringa5m/sec 2 (equivalent to traveling on aninclined surfaceof30 o ). The results indicate that legs1and 2 losecontact at inclinationangles larger than 78 o (shownas anegativefriction force). However, the lateralforceis larger than the friction limit for all inclination angles, resultinginlateral slippage for any internalconfiguration.Inorder to completea stable motionat a5m/sec 2 the inclinationof the “front”leg (leg3) must be reduced,as showninFig.7b.This changeadjusts the forcedistribution, sim ilar toh umans clim bin ga s teep hi ll. The ne w in ternalcon fi guration en ab les stab le mo tio nas long as in cli natio nangl ef or le g 3 is in t he rang eo f68 o -5 3 o ,and legs1and 2 arelarger than 78 o . Legs 1 and 2 -40 -2 0 0 20 40 60 80 100 020406080100 n [deg] Force [N] Fd Fl Fu (a)(b) Fig.7. Accelerated motion of 5m/sec 2 . Sim ilar resu lts areo btain ed fo r circularmo tio n. Again , s ymmetricin ternal con figuration r esu lts eithe r in tip over or im me di ate s lip page in one or mo rel eg s. Asymmetricinternalconfiguration enables the robot to safely complete the required motioneven for sharpcurvatures with relatively high speeds. Travel Direction 330 S.Shovaland M.Shoham To verify the simulation results wehaveconducted fieldexperiments usingour inflatable mobile platform showninFig. 1. In theseexperiments wemeasured the stab ility li mits fo r vario us internal. The r esu lts ind i cateclo sem atchbetween t he theoretical simulations and the field experiments.For example, stability limit for horizontal s ur face with symmetricconfiguration (showni nFig s.5 -6)is obtaine d for inclinationangle of 60 o ,compared with53 o determinedin the simulation. Motion over inclined surfaceof 30 o is stable wheninclinationangle for leg3 is in the range of 65 o -55 o compared with 68 o -53 o determinedin the simulation. 5Field Experiments 6Conclusions Anew designfor aparallelmobile robot is presented.The robot consists of three legs,eachdriven by anasynchronous mechanismconnected to the legs witha sphericaljoint.Eachleg is connected toanupper platform withrevolutejoint and additionalencoders,measuring the revoluteangle of the upper joints.These encoders providedata used bythe kinematicmodel forearly detectionand correctionofpositioning errors generated by odometry.Early detectionand correction of odometricerrors in eachleg prevent accumulation of significant errors of the upper plate,and canidentify irregularities on the surface. A simplified dynamicmodelprovides a solutionfor the forces applied on the robot.The mo del determi ne s whether a s pe cif ic t askcanbe r el iab ly pe rfo rme d, given a specific surface topography and friction. The model canalsodetect instabilities either by loosing contact with the ground (tipover),or by slippage (longitudinalor lateral).Anunstable configuration canbeavoidedbychanging the inclination angles between the legs and the upper plate. This featureallows the robot tocompletemotions in complex terrains whereconventional robots cannot maintain stability due toinertialforces, surface topography,or friction constraints. References 1. Ben Horin (Dombiak) P.,1999,Analysis and Synthesis of anInflatable Parallel Robot”, M.Sc.Thesis,Technion, Haifa. 2.Ben Horin (Dombiak) P.,Shoham,M.,and Grossman,G.,“AParallel Six Degrees of- FreedomInflatable Robot,” ASME 2000 Mechanismand Robotics Conference, Washington, 2000. 3.Ben Horin R.,“Criteriafor Analysis of Parallel Robots”, D.Sc.Thesis,Technion, Haifa, 1997. 4. DoW.Q.D.,Yang D.C.H.,“InverseDynamics Analysisand simulation of a PlatformType of Robot”, JournalofRoboticSystems, Vol. 5,No. 3,pp. 209-227,1998. 5. Hunt K.H.,1983,“StructuralKinematics of In-Parallel-Actuated Robot Arm”, ASME JournalofMechanisms Transmissionsand automation in Design, Vol. 105,pp. 705- 712. Motion Analys is of aParallel Mobile Robot 3 31 6.ShovalS.and ShohamM.“ARedundant Parallel Mobile Mechanism”, Proceedings of IEEEInternationalConferencefor Robotics and Automation, May 2001,Seoul,Korea, pp. 2273-2279. 7.Stewart D.,1965,“Aplatform with six Degrees of Freedom”, Proceedings of Instittute of Mechanica lEngineering ,L on don England , Vol .180,pp . 371- 386. 8. TahmsebiF.,TsaiL.W.,1994,“Closed-FormDirect Kinematics Solution of aNew Parallel Minimanipulator”, Transactionsof the ASME, Vol. 116,pp. 1141-1147. 9. TsaiL.W.,TahmasebiF.,1983,“Synthesis and Analysis of aNew Class of Six Degree-of-Freedom ParallelMinimanipulators”, JournalofRoboticResearch, Vol. 10, pp. 561-580. 1 2 2 1 2 [...]... different handling conditions While the proposed static characterizations of vehicle lateral stability are not appropriate as instantaneous measures of vehicle stability they do highlight the importance of considering the height of the cg and the vehicle S Yuta et al (Eds.): Field and Service Robotics, STAR 24, pp 343–354, 2006 © Springer-Verlag Berlin Heidelberg 2006 344 A Diaz-Calderon and A Kelly... IEEE Int.Conf.on Robotics and Automation, Detroit, USA, pp 1226– 1233, 1999 7 T.G.Sugar, V.Kumar, “Control of Cooperating Mobile Manipulators,” IEEE Transactions on Robotics and Automation, VOL.18, NO.1, pp 94 103 , 2002 8 K.Kosuge, T.Oosumi, M.Sato, et al, “Transportation of a Single Object by Two Decentralized-Controlled Nonholonomic Mobile Robots,” Proc of 1998 IEEE Int Conf Robotics and Automation,... measure of the proximity of the vehicle to a tipover event as a function of motion and configuration states The proposed stability algorithm consists of two parts: 1) an estimation and prediction system and 2) a dynamic module The estimation and prediction system estimates the current and future states of the vehicle using on-board sensors to measure relevant dynamic quantities The dynamic module computes... vector describes the motion of the cg of the vehicle as well as the vehicle attitude: x = v a ω z αz φ θ T (1) where v and a are the linear velocities and accelerations of the cg, and ω z and αz are the angular velocities and accelerations of the cg about the vehicle’s vertical, and φ and θ are the Euler angles that describe the vehicle attitude The system dynamics will be described by the following... specific force (t), and roll (Φ) and pitch (Θ) inclinometers z = zω zδ zvx zvz zta zΦ zΘ T (7) Eq 8 gives the relationship between the state (x) and measurement (z) vectors, which is non-linear The measurement Jacobian is defined as H = ∂ h and the matrix ∂x i B R = Rz (0) Ry (θ) Rx (φ) is the ZYX-Euler angles rotation matrix that describes the orientation of the truck relative to the earth, and B R = i R−1... J.S.Bay, “Toward the Development of a Material Transport System using Swarms of Ant-like Robots,” Proc of 1993 IEEE Int Conf Robotics and Automation, Atlanta, USA, pp 766–771, 1993 2 L.Chaimowicz, T.Sugar,V.Kumar, et al, “An architecture for Tightly Coupled Multi-Robot Cooperation,” Proc.of 2001 IEEE Int Conf Robotics and Automation, Seoul, Korea, pp 2992–2997, 2001 3 K.Inoue, T.Nakajima, “Cooperative... The results are shown in Figs 10 11 Fig 10 shows the pathes of the leader controlled by the human operator and the follower using the proposed algorithm respectively and the snap shot of the experiments Fig 11 shows the translational and angular velocity of the leader It is evident that the leader realizes the motion instructed by the joystick input As shown in Figs 10 11, the transportation task... according to the force direction applied to the robot When the robot is pushed forward, the offset is set equal to Cof f (> 0)(Fig 2(b-1)), and when the robot is pulled backward, the offset is set equal to −Cof f (Fig 2(b-2)).We refer to this caster-like dynamics as the dual-caster action 3 Function Allocation In this research, the task is to control two tracked mobile robots which is transporting a relative... coming close to the object or itself and generates the avoidance motion Then we can see that the leader and the follower could execute the transportation task successfully Teleoperation System for Two Tracked Mobile Robots 341 Fig 10 Experimental results: real trajectory of the leader controlled by the human operator and the follower using the proposed algorithm(above) and the view of the experiments(below)... should have two functions, transporting the object in coordination and avoiding collision between obstacles and the transportation system including robots and the object In this research, we allocate the two types of function to each robot One is to control the position of the object and the other is to control the orientation of the object and to transport the object in coordination We allocate the position . robot ,and section4 shows simulation results for motions in various configurations and con straints.Section5 verifies the simulationresults withexperiments conducted withour prototype platform and. units.The mechanismis designed for applications that requirealightweight and easy S. Yuta et al. (Eds.): Field and Service Robotics, STAR 24, pp. 323–331, 2006. © Springer-Verlag Berlin Heidelberg 2006 32 4S. Shovaland M.Shoham depl oyable . MechanicalEngineering, Technion,Haifa, Israel, shoham@tx.technion.ac.il Abstract. This paper presents akinematicand forceanalysis of amobile robot built on the principle of parallel mechanisms.The robot consists of an upper plateconnected