36 2M athematical Mo del Construction of aM ec hatronic Serv oS ystem requiredallowable error. Whenconstructing the mo del,inorder to obtai nthe simplemodel with satisfying the required precision, the model should be the lowspeed 1st order model for the lowspeed operation. Additionally,inthe middle speedoperation from1/20 to 1/5 of ratedspeed, the evaluationerror of thelow speed 1st order model is bigger than the required allowance error andsmaller than that in the middle speed 2nd order model. In the high-speed motionover1/5 of ratedspeed of the motor, the evaluationerrorbetween the lowspeed 1st order model and middle speed 2nd order model is bigger than the required allowance error. From theseresults, the adaptable scale and bo undary of ther educed order mo del can be judged. The correctm od eling of actual industrial mec hatronic serv os ystem by derivedreduced order model wasverified by experiment. The adopted ex- perimentaldevice for verification is aDEC-1similar to item 2.1.3 (refer to experimental deviceE.1). The lowspeed of motionvelocityis5[rad/s] about 1/20 of rated speed, and middle speed is 20[rad/s] about 1/5ofratedspeed. Fig. 2.9 illustrates the modelingerrorbetween the outputand the reduced order model in the experiment. From theresults in Fig. 2.9,inthe lowspeed operation, the modelingerrorofboth thelow speed 1st order model and the middle speed 2nd order model is smaller than 0.05[rad], whichisalmost con- sistentwith the experimental results. In the middle speed operation, the error between the lowspeed 1st order model and experimental results is bigger than the maximal0.14[rad]. In the middle speed 2nd order model, the modeling error is smaller than 0.05[rad]. Therefore, the modelingisappropriate.From these experimentalresults, the appropriateness of the reduced order model expressing the dynamic of industr ial mechatronic servosystem wasverified. 00. 20.4 0 0 .1 T ime[ s ] M odeling e rro r [ r a d ] L o w speed model M iddle speed model 00. 20.4 0 0 .1 T ime[ s ] M odeling e rro r [ r a d ] L o w speed model M iddle speed model (a) Lo ws pe ed (5[rad/s]) (b) Middle sp eed (20[rad/s]) Fig. 2.9. Evaluation of lowspeed 1st order model and middle speed 2nd order model 2.3L inear Mo del of the Wo rking Co ordinates of an Articulated Rob ot Arm 37 Objec t i v e tra jec t o ry [Wo r king c oor din a t e ] [Wo r king c oor dina t e ][Joint c oor dina t e ] M o t o r I n v e rse kinema t i c s K inema t i c s S e rvo c ontroller Objec t i v ejoint a ngle F ollow ing joint a ngle F ollow ing tra jec t o ry D i v i s ion b y r efer enc einput t ime int e rva l Fig. 2.10. Block diagram of industrial articulated robot arm 2.3 Linear Mo del of the WorkingCoordinates of an Articulated Rob ot Arm In an industrialarticulatedrobot arm, instructions aregiven in working co- ordinate.The motorisdriven in the jointcoordinate space transformedby nonlinearcoordinatesbycalculation in thecontroller.Hence, the mechanism part is movedinthe working coor dinate space.Therefore, according to the specialregioninworking coordinates, there is the problemofprecisiondete- riorationofthe contourcontrol of robotarm. The approximation model (2.46) in theworking coordinate of an articu- lated robotarm an dits approximationerror(2.54) arederived. By usingthis model,the working linearizable approximation possible re- gionfor keepingthe movementprecisionofanarticulatedrobot armwithin the allowance is clarified.The region, in which the high-precision contour con trol of the rob ot arm is capable to realize, is confirmed. Besides, from the discussion in this section, by holding thisv iew of approx imatione rror, the one axischaracteristic in the jointcoordinate given in 2.1 and2.2 can express the ch aracteristics of them ec hatronic serv os ystem in wo rking co ordinates. The simplification of thea nalysisa nd designo fm ec hatronic serv os ystems is ve ry important. 2.3.1 AWorking Linearized Model of an Articulated RobotArm (1)AnIndustrial ArticulatedRobot Arm Control System Theblock diagram of contourcontrol of an industrialarticulatedrobot armis illustrated in Fig. 2.10. At first,the objective trajectory in working coordinates is dividedi nt oe ac hr eference input time in terv al (refer to section 3.2 and 3.3). The jointangle of eachaxis is calculated at eachdivision point. The rotationangle of theservomotor is controlled by various axisjointangles with constantvelocitymovements based on the objectivejointangle dividedin jointcoordinate.The servomotor of eachaxis is rotated only with its defined movement. Thus, the arm tip is movedalong the objectivetrajectory of the working coordinate with the coordinate transform in the armmechanism. If the objectivetrajectory is given in working coordinatesand the robot armcontrol of eachaxis is independentofthe jointcoordinate with nonlinear 38 2M athematical Mo del Construction of aM ec hatronic Serv oS ystem transform, thefollowing trajectory is evaluated in working coordinateswith nonlinear transform. When controlling arobot armwith this control pattern, the control system of an ind ustr ial robot arm, with eachlinear independentco- ordinate axis, is generallyapproximatedinworking coordinates. Forpreparing the discussion (in2.3.2)ofappropriate linearapproximation in this working coordinate,the working linearized approximation trajectory,basedonthe ac- tual trajectory and working linearized model of working coordinate of this robotarm controlsystem, is derived. (2)Actual Tr ajectory of aTwo-Axis Robot Arm Foranalyzing the characteris tics of multiple axes, the natureoftwo axes is discussedand the analysisisexpanded into multiple axes in 2.3.2(4). In Fig. 2.11, two rigid links ar eexpressed with .The conceptualgraph of atwo- axisrobot armwith movementofthe tiponthis plate is shown. The ( θ 1 ,θ 2 ) in figureisthe jointangle in jointcoordinates. ( p x ,p y )isthe tipposition in working coordinates, l 1 , l 2 arethe lengths of axi s1andaxis 2, respectively. This two-axis robot arm is the basic structureofamulti-axis robot arm. In the SCARA robot arm, the plate position determination is carried out for these twoaxes. At first,for determining the relationship between the working coordinate andjointcoordinate,the transformation fromjointcoordinate ( θ 1 ,θ 2 )towork- ingcoordinate ( p x ,p y )(kinematics) and the transformation fr om working co- ordinate ( p x ,p y )tojointcoordinate ( θ 1 ,θ 2 )are explained. From Fig. 2.11, the kinematics is as p x = l 1 cos θ 1 + l 2 cos( θ 1 + θ 2 )(2.38a ) p y = l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ) . (2.38 b ) x y θ ( , ) l l 1 2 θ x y pp 1 2 J oint J oint link 1 link 2 Fig. 2.11. Structure of two-degree-of-freedomarticulated robot arm 2.3L inear Mo del of the Wo rking Co ordinates of an Articulated Rob ot Arm 39 - K p + 1 - s Objec t i v ejoint a ngle F ollow ing joint a ngle [Joint c oor dina t e ] M o t o r S e rvo c ontroller P o s i t ion loop Fig. 2.12. Block diagram of 1st order model in jointcoordinate of industrial mecha- tronic servosystem From thesolution of ( θ 1 ,θ 2 )inequation (2.38 a )and (2.38 b ), the inverse kine- matics is given as θ 1 =sin − 1 ⎛ ⎝ p y  p 2 x + p 2 y ⎞ ⎠ − sin − 1 ⎛ ⎝ l 2 sin θ 2  p 2 x + p 2 y ⎞ ⎠ (2.39 a ) θ 2 = ± cos − 1  p 2 x + p 2 y − l 2 1 − l 2 2 2 l 1 l 2  (2.39 b ) wherethe symbol of equation (2.39b )denotes that one assigned pointinwork- ingcoordinate hastwo possibilities in the jointcoordinate. Next,the dynamics of therobot armisgiven in the jointcoordinate.Inan industrial robot arm, if the gearratioislarge, then theload inertiaissmall. Moreover, if using aparallel link, the effectofno-angle part of inertiamatrix is small.The servomotor in theactuator performs the controlonthe robot armineachindependent axis. Foranactual industrial robot arm, when the motionv elo cit yo ft he robo ta rm is be lo w1 /20 of rateds pe ed, eac ha xis can be expressed with a1st order system as (refer to 2.2.3). dθ 1 ( t ) dt = − K p θ 1 ( t )+K p u 1 ( t )(2.40a ) dθ 2 ( t ) dt = − K p θ 2 ( t )+K p u 2 ( t ) . (2.40 b ) The model expressed by equation (2.40) is called ajointlinearized model. Here, u 1 ( t )and u 2 ( t )denotes the angle input of axis 1and axis2,respec- tively. K p denotes K p 1 of equation (2.23) in thel ow sp eed 1st order mo del of 2.2.3. Fig. 2.12i llustrates the blo ck diagram of the1 st order system. In this section, eachaxis dynamic is expressed by equation (2.40) in jointcoordinates. Forclarifyingthe expression of actualrobot dynamics by the jointlinearized model. The following discussion is carried out with this assumption. The robot arm is analyzed about howtotrace the objective trajectory divided into small intervals. Concerning the various trajectoriesdivided from the ob jectiv et ra jectory ,t he be ginning po in ta nd end po in ti nw orking co ordi- nateswithin one divided small interval are expressed by ( p 0 x ,p 0 y ), ( p ∆T x ,p ∆T y ), 40 2M athematical Mo del Construction of aM ec hatronic Serv oS ystem ( , ) p pp ( , ) x y θ θ TT ∆ ∆ θ θ 0 0 1 2 1 2 x y p 00 x y ∆∆ T T Fig. 2.13. One in terv al of ob jectiv et ra jectory divided by reference input time interval respectively,and the beginning pointand end pointinjointcoordinatesare expressedby(θ 0 1 ,θ 0 2 ), ( θ ∆T 1 ,θ ∆T 2 ), respectively.The relationship between joint coordinates and working coordinatesinthis small interval is given in Fig. 2.13. Therelationbetween ( p 0 x ,p 0 y )and ( θ 0 1 ,θ 0 2 )aswell as between ( θ ∆T 1 ,θ ∆T 2 )and ( p ∆T x ,p ∆T y )are expressedasbelowbasedonthe expression of therelationship between working coordinatesand jointcoordinatesfromequation (2.38 a )and (2.38 b ). p 0 x = l 1 cos θ 0 1 + l 2 cos( θ 0 1 + θ 0 2 )(2.41a ) p 0 y = l 1 sin θ 0 1 + l 2 sin(θ 0 1 + θ 0 2 )(2.41b ) p ∆T x = l 1 cos θ ∆T 1 + l 2 cos( θ ∆T 1 + θ ∆T 2 )(2.41c ) p ∆T y = l 1 sin θ ∆T 1 + l 2 sin(θ ∆T 1 + θ ∆T 2 ) . (2.41 d ) Concerning the industrial robot arm, from the given constantangle ve- locityinput ( v 1 ,v 2 )ofeachaxis in divided small intervals, the angle in- put(u 1 ( t ) ,u 2 ( t )) for eachaxis dynamic of the robot arm (2.40) is given as ( u 1 ( t ) ,u 2 ( t )) u 1 ( t )=θ 0 1 + v 1 t, v 1 = θ ∆T 1 − θ 0 1 ∆T (2.42a ) u 2 ( t )=θ 0 2 + v 2 t, v 2 = θ ∆T 2 − θ 0 2 ∆T (2.42b ) where ∆T denotesthe referenceinput time interval (refer to 3.2, 3.3).The time of thebeginningdivision pointiszero. If the angle input is expressed by equation (2.42), the robot arm position in working co ordinatescan be derived. Whenthe objective trajectory is the 2.3L inear Mo del of the Wo rking Co ordinates of an Articulated Rob ot Arm 41 same as theposition of theactual trajectory as ( θ 1 (0),θ 2 (0)) =(θ 0 1 ,θ 0 2 )inthe initial time of robotarm, the position in jointcoordinatesofthe robotarm is as belowfromthe solutionofdifferential equation after putting angle input of equation (2.42) into (2.40) (refer to app end ix A.2). θ 1 ( t )=θ 0 1 + v 1 λ ( t )(2.43a ) θ 2 ( t )=θ 0 2 + v 2 λ ( t )(2.43b ) λ ( t )=t + e − K p t − 1 K p . (2.44) At this time, the po sition of the rob ot arm in wo rking co ordinate canb e calculated when putting the nonlinear transform equation (2.43) into (2.38 a ), (2.38 b ) p x ( t )=l 1 cos  θ 0 1 + v 1 λ ( t )  + l 2 cos  θ 0 1 + θ 0 2 +(v 1 + v 2 ) λ ( t )  (2.45 a ) p y ( t )=l 1 sin  θ 0 1 + v 1 λ ( t )  + l 2 sin  θ 0 1 + θ 0 2 +(v 1 + v 2 ) λ ( t )  . (2.45 b ) This equation (2.45) expresses the actual trajectory of the robot arm tip in working coordinates. Concerning thisactual trajectory,asthe problemof this section, the working linearized approximation trajectory in the working linearized model is derivedafterlinearized approximationofeachcoordinate axisindependently of the working coordinates. (3) Working LinearizedApproximationTrajectory of aTwo-Axis Rob ot Arm In working coordinates, the controlsystem of the robot arm is as belowwhen x axis y axisa re linearly appro ximatedi ndep endent ly ,r esp ectiv ely d ˆp x ( t ) dt = − K p ˆp x ( t )+K p u x ( t )(2.46a ) d ˆp y ( t ) dt = − K p ˆp y ( t )+K p u y ( t )(2.46b ) where(ˆp x ( t ) , ˆp y ( t )) denotes the rob ot arm po sition in the wo rking co ordinate linearly appro ximation.( u x ( t ) ,u y ( t )) denotes the position input in working coordinates. Thisequation (2.46) is the working linearized mo del as thediscus- sion object of this section. When the objectivetrajectory is dividedasshown in Fig. 2.13 with thelinearized approximationequation (2.46), the robot arm resp onse at small intervals is derived. Here, the objectivetrajectory is the same as theposition of theworking linearized approximation trajectory as (ˆp x (0), ˆp y (0)) =(p 0 x ,p 0 y )atthe initial time of therobot arm. Strictly speak- ing,The input in working coordinate corresponding to the input (2.42) in thejointcoordinate needs to be derivedaccordingtothe coordinate trans- form (2.38 a ), (2.38 b ). If the input in the wo rking co ordinate is nota constan t 42 2M athematical Mo del Construction of aM ec hatronic Serv oS ystem velocity, the input in working coordinatesisapproximatedwith acon stant velocityby u x ( t )=p 0 x + v x t, v x = p ∆T x − p 0 x ∆T (2.47a ) u y ( t )=p 0 y + v y t, v y = p ∆T y − p 0 y ∆T . (2.47 b ) Its approx imatione rrorc an almost be neglected. If the input of thee quation (2.47) is puti nt ot he wo rking linearized mo del of equation (2.46), thew orking linearized appro ximation traj ectory of the rob ot arm from the solution of differential equationisas ˆp x ( t )=p 0 x + v x λ ( t )(2.48a ) ˆp y ( t )=p 0 y + v y λ ( t ) . (2.48 b ) That is, the working linearized approximation trajectory correspondingtothe actualtrajectory (2.45) of therobot arminworking coordinatesisgiven by equation (2.48). 2.3.2Derivation of Adaptable Region of the WorkingLinearized Model (1) Approximation Error of the WorkingLinear ized Model Fr om thec omparison be twe en the actual tra jectory (2.45) of the rob ot arm control system and the working linearized approximation trajectory (2.48), the approx imationp recisiono ft he wo rking linearized mo del fort he ob ject discussedi nt his sectioni se va luated. The approx imatione rrori nt he wo rking coordinate is the errorbetween equation (2.45) and (2.48) as e x ( t )=ˆp x ( t ) − p x ( t )(2.49a ) e y ( t )=ˆp y ( t ) − p y ( t ) . (2.49 b ) ( e x ( t ) ,e y ( t )) of equation (2.49) is called the working linearized approximation error. In order to evaluate separately the item aboutthe time andthe item ab outthe space in equation (2.49), theactual position of therobot armin working coordinatesexpr essed by equation (2.45) is calculated as belowwith 1st order approximationbyTaylor expansion when the movementof(θ 0 1 ,θ 0 2 ) is very small. ˜p x ( t )=l 1 { cos( θ 0 1 ) − sin(θ 0 1 ) v 1 λ ( t ) } + l 2 { cos( θ 0 1 + θ 0 2 ) − sin(θ 0 1 + θ 0 2 )(v 1 + v 2 ) λ ( t ) } (2.50 a ) ˜ p y ( t )=l 1 { sin(θ 0 1 )+cos( θ 0 1 ) v 1 λ ( t ) } + l 2 { sin( θ 0 1 + θ 0 2 )+cos( θ 0 1 + θ 0 2 )(v 1 + v 2 ) λ ( t ) } . (2.50 b ) 2.3L inear Mo del of the Wo rking Co ordinates of an Articulated Rob ot Arm 43 Between the actual trajectory and the 1st order appr oximationtrajectory by Taylor expansion of equation (2.50) is as [9] p x ( t )=˜p x ( t )+l 1 o { v 1 λ ( t ) } + l 2 o { ( v 1 + v 2 ) λ ( t ) } =˜p x ( t )+o { λ ( t ) } (2.51 a ) p y ( t )=˜p y ( t )+l 1 o { v 1 λ ( t ) } + l 2 o { ( v 1 + v 2 ) λ ( t ) } =˜p y ( t )+o { λ ( t ) } . (2.51 b ) The o { λ ( t ) } in equation (2.51) denotes the high level infi nitesimal of λ ( t ). By usingtriangle inequality,the size of errorbetween the actual trajectory and the working linearized approximation trajectory can be restrainedbyequation (2.48) and (2.50) | ˆp x ( t ) − p x ( t ) |≤|ˆp x ( t ) − ˜p x ( t ) | + | ˜p x ( t ) − p x ( t ) | = | ε x λ ( t ) | + | o { λ ( t ) }| (2.52a ) | ˆp y ( t ) − p y ( t ) |≤|ˆp y ( t ) − ˜p y ( t ) | + | ˜p y ( t ) − p y ( t ) | = | ε y λ ( t ) | + | o { λ ( t ) }| (2.52b ) where(ε x ,ε y )is ε x = v x + p 0 y v 1 + l 2 sin(θ 0 1 + θ 0 2 ) v 2 (2.53 a ) ε y = v y − p 0 x v 1 − l 2 cos( θ 0 1 + θ 0 2 ) v 2 . (2.53 b ) If the po sition of the rob ot arm is dep ended on ve lo cit y, thereh as thee rror item notd ep ended on the time. When λ ( t )i sv ery small, the item o { λ ( t ) } in equation (2.51) can be neglected. Therefore, the working linearized approxi- mation errorc an be appro ximateda s e x ( t ) ≈ ε x λ ( t )(2.54a ) e y ( t ) ≈ ε y λ ( t ) . (2.54 b ) That is, if the λ ( t )can be very small andthe divisioninterval of the ob- jectiv et ra jectory is ve ry small,t he wo rking linearized appro ximation error canbeexpressed by equation (2.54). The equation (2.54) is given by item ( ε x ,ε y )d ep ended on the rob ot arm po sition in equation (2.53) and the in- tegral with item λ ( t )dependentontime. The ( ε x ,ε y )inequation (2.53) is the function of the robot arm position ( p 0 x ,p 0 y ), ( θ 0 1 ,θ 0 2 )a nd motion ve lo c- ity(v x ,v y ), ( v 1 ,v 2 ). Here, the robot arm position ( θ 0 1 ,θ 0 2 )expressed in joint coordinates can be expressed in working coordinatesbykinematic equation (2.38 a ), (2.38 b ). Moreover, the motion velocityinjointcoordinates, expressed by ( v 1 ,v 2 )=(( θ ∆T 1 − θ 0 1 ) /∆T, ( θ ∆T 2 − θ 0 2 ) /∆T )inequation (2.42), can be alsoexpressed in working coordinatesas(p 0 x ,p 0 y ), ( p ∆T x ,p ∆T y )fromkinematics (2.38 a ), (2.38 b ). Equation(2.54) can expressthe robotarm position ( p 0 x ,p 0 y ), ( p ∆T x ,p ∆T y )i nw orking co ordinates. Thise quation (2.54) expresses the wo rking linearized approximation error, as thepurpose.Fromthe evaluating the size 44 2M athematical Mo del Construction of aM ec hatronic Serv oS ystem of this error, the appropriation of theworking linearized model of th econtrol system of therobot armaswell as the working linearizable approximation possible region can be derived. (2) QuantityEvaluation of the WorkingLinearized Model Thes mall region of wo rking linearized appro ximation erroro ft he wo rking linearized mo del as (2.46) in wo rking co ordinateso fr ob ot arm, i.e.,w ork- ingl inearizable re gion, is quan titativ ely ev aluated. In Fig. 2.14, within the mo ve able regiono ft he robo ta rm is enclosedb ya dotted line in wo rking co ordinates, whent he robo ta rm is mo ve da long the arro wd irection from eachbeginningpoint(p 0 x ,p 0 y )(bullet • in figure) of 188points dividedineach 0.2[m],the value of ( ε x ,ε y )about position of working linearized approxima- tion erroriscalculatedby(2.53) (linefrombullet • in figure) and its results are illustrated. The length of the arm is l 1 =0. 7[m], l 2 =0. 9[m]. Themotion velocityis v x =0. 1[m/s], v y =0. 1[m/s].The symbol of inverse kinematics (2. 39b )ofthe robotarm is oftenpositive.FromFig. 2.14, the approximation precision of the working linearized model deterioratesnear the boundary of themoveable regionalong the motiondirection of the robot arm. Moreover, in the shrinking regionofthe robotarm, the working linearized approxima- tion errorbecomes large. Since the working linearized approximation er roris dependent on theposture of thearm but absol utely independentonthe posi- tion in wo rking co ordinateso ft he arm, ther esults of the wo rking linearized approximation errorinFig. 2.14expresses that, the robot arm is movednot only alongthe errordirection, butalso rotated around the original pointin Fig. 2.14a long an yd irection, and also the mo ve men td irection of arm is along − 22 − 2 2 x [ m ] y [ m ] ε x 0. 0 1 [ m /s] ε y 0. 0 1 [ m /s] M o v ing dir e c t ion Fig. 2.14. Working linearized approximation error for various initial position (bullet • :initial position of robot arm; division from bullet • :working linearized approxi- mation error vector ( ε x ,ε y )) 2.3L inear Mo del of the Wo rking Co ordinates of an Articulated Rob ot Arm 45 thearrowdirection in the figureand it is the dependent item of the working linearized approximation error. Next,when changingthe view point, from one beginning pointofthe robot arm(the distance from the initial pointtothe armtip position is written as r =  ( p 0 x ) 2 +(p 0 y ) 2 ), howthe working linearized approximation er rorchanges alongvariousmotion directions canbeseen. At four points r =0. 25, 0.38, 1.5, 1.55[m] an dwith motionvelocity v =  v 2 x + v 2 y = √ 0 . 02 ≈ 0 . 141[m/s], when thearm is movedone cycle2π at eachdirection with regardinginitial position as the center, the results of position dependentitem size  ε 2 x + ε 2 y of theworking linearized approximation errorare illustratedinFig. 2.15. The horizontal axis φ of Fig. 2.15 representsthe movementangle of arm. From the angle standard φ =0[rad] of angle stretchingdirection, φ = π [rad] denotes the arm shr inkin gdirection. From Fig. 2.15, at r =0. 25[m]and 1.55[m] near the boundary of thearm moveable region(0 . 2 ≤ r ≤ 1 . 6[m]),the working linearized approximation errorbecomes largeatthe armstretchingaction. In the movementatthe pull-pushdirection and verticaldirection, the working linearized approximation errorbecomes fairly small. When the working linearized approximation error(2.54) is dependent on time,the time shift with K p =15[1/s]ofthe time depending item λ ( t ), is illustrated in Fig. 2.16. In the reference input time interval ∆T =0. 02[s], λ ( t )is0.0027[s]. From Fig. 2.15, theposition dependent item size  ε 2 x + ε 2 y of theworking linearized approximation errorisbelow0.001[m/s] with any direction motionwithin the region 0 . 38 ≤ r ≤ 1 . 5[m]. Therefore, themaximum of theworking linearized approximation erroris0.0027[mm]. This value is about 0.1%ofthe small interval length 0 . 141[m/ s] × 0 . 02[s]=0. 00282[m]with reference input time interval ∆T =0. 02[s]and it is very small value. That is, when the reference input time interval is 0.02[s] with the robot arm motion velocity0.141[m/s], the working linearized approximation erroriswithin 0.1% 0 0 0 . 001 0 . 002 0 . 003 φ [ r a d ] ε x 2 + ε y 2 [ m /s] π 2 π r = 0 . 2 5 [ m ] r = 0 . 3 8 [ m ] r =1.5[ m ] r =1.55[ m ] Fig. 2.15. Working linearized approximation error for various movementdirection φ ,initial position of robot arm r ( r =0. 25[m], r =0. 38[m], r =1. 5[m], r =1. 55[m]) [...]... Mathematical Model Construction of a Mechatronic Servo System x Tip 3rd axis y Joint Fig 2.17 Three-degree-of-freedom robot arm x axis and the third axis (4) Expansion to a Multi-Axis Robot Arm In the discussion so far, the working linearizable region for a two-axis robot arm is derived In this part, the working linearizable region is discussed from a two-axis robot arm to a multi-axis robot arm Concerning the... linearizable region of a two-axis robot arm along the z axis direction Since the 4th axis is the self-rotation of the end-effect, there is no need to make operation linearizable approximation Next, when determining the position in the working coordinates of a sixaxis robot arm, three axes are considered from the base The third axis of this six-axis robot arm is adopted as the y axis of a two-axis robot arm expressed... third axis and x axis is illustrated by Fig 2.17 When making a linearizable approximation in this plate by this third axis and x axis, it is different from the linear approximation of a two-axis robot arm discussed in the previous part The former is that one axis is rotated and one axis is moved directly The latter is that both axes are rotated That is to say, the linear approximation of the two-axis robot... previous part means that the transformation from two axes rotation to two axes direct movement is possible Therefore, the transformation from one axis rotation and one axis direct movement in the formed plate by the third axis and x axis is same as the linear approximation discussion of the two-axis robot arm discussed in the former part That is to say, the robot arm in the formed plate by the third axis and. .. coordinates The working linearizable region of the three-axis robot arm is the region that the working linearizable region of the two-axis robot arm is rotated by the y axis Considering the third axis similar to hand, it is no need to make the operation linear approximation for self-rotation of the end-effect at the ball surface regarding the hand tip as the center in the operation space ... coordinate velocity (vx , vy ) = (v cos φ, v sin φ) ∆T ∆T 0 and the motion velocity in joint coordinate (v1 , v2 ) = ((θ1 −θ1 )/∆T, (θ2 − 0 θ2 )/∆T ) are calculated 4 Using equation (2.53), the position dependent item of the working linearized approximation error (εx , εy ) is calculated And its size ε2 + ε2 x y is calculated 5 Using equation (2 .44 ), the time dependent item of the working linearized approximation... coordinate and joint coordinate are respectively (u0 , u0 ), (u0 , u0 ) The actual trajectory in working coordinates 2 1 y x 0 0 and joint coordinates are respectively (p0 , p0 ), (θ1 , θ2 ) The position of the x y working linearized approximation trajectory in working coordinate is (ˆ0 , p0 ) px ˆy When we put equation (2 .42 ) into (2 .40 ), solve (θ1 (t), θ2 (t)) by using initial 0 0 condition (θ1 , θ2 ) and. .. we put equation (2 .47 ) into (2 .46 ) and solve (ˆx (t), py (t)) by using the p ˆ initial condition (ˆ0 , p0 ), the working linearized approximation trajectory is px ˆy calculated by px (t) = p0 + (u0 − p0 )σ(t) + vx λ(t) ˆ ˆx ˆx x (2.57a) py (t) = ˆ (2.57b) p0 ˆy + (u0 y − p0 )σ(t) ˆy + vy λ(t) From the error between equation (2.57) and (2.55), the error between the actual trajectory and the working linearized.. .46 2 Mathematical Model Construction of a Mechatronic Servo System δ(t) [s] 0.002 0.001 0 0 0.01 Time [s] 0.02 Fig 2.16 Time dependence of working linearized approximation error λ(t) of the objective trajectory in one division scale of objective trajectory, and the working linearizable approximation possible region can be as 0.38... initial time and the working linearized x y approximation trajectory (ˆ0 , p0 ) The second item is the working linearized px ˆy approximation error (2. 54) derived when the objective trajectory in 2.3.2(1) is identical with the actual trajectory and the working linearized approximation trajectory The third item is the error item based on the position difference between the objective trajectory and the actual . arm, instructions aregiven in working co- ordinate.The motorisdriven in the jointcoordinate space transformedby nonlinearcoordinatesbycalculation in thecontroller.Hence, the mechanism part is movedinthe. the ch aracteristics of them ec hatronic serv os ystem in wo rking co ordinates. The simplification of thea nalysisa nd designo fm ec hatronic serv os ystems is ve ry important. 2.3.1 AWorking Linearized Model. working coordinatesand the robot armcontrol of eachaxis is independentofthe jointcoordinate with nonlinear 38 2M athematical Mo del Construction of aM ec hatronic Serv oS ystem transform, thefollowing