Massachusetts Institute of Technology Department of Mechanical Engineering
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1 cm cm Poblem Massachusetts Institute of echnolog Depatment of Mechanical Engineeing. Intoduction to obotics Sample Poblems and Solutions fo the Mid-em Exam Figue shows a obotic vehicle having two poweed wheels. he angula velocit of the ight wheel is and that of the left wheel is L. he distance between the wheels is cm and the cente of the axle is at point O v. he tie adius measues cm. cm. Answe the following questions. Y L Y ajecto of Ov v = 6.8 cm/s 6 cm D Ov X A B C cm O X Figue Figue Y Ov L cm E End effecto X Y cm ajecto of end effecto F 5 cm E v Figue a). he axle cente Ov tacked the path shown in Figue. It stated at the oigin of the wold coodinate sstem O-XY, moved along a cicle of adius cm centeed at point A, and then moved along anothe cicle of adius 6 cm fom point B to D. he tacking speed was v 6.8 cm/s thoughout the tajecto. Obtain the angula velocities of both wheels at each O cm Figue X
2 cm Angula velocit, ad/sec segment of the tajecto. Plot D? and L against time. How long does it take to move fom O to b). he vehicle caies an instument at the endpoint E, which is cm awa fom the axle as shown in Figue. Figue shows a cicula tajecto of adius 5 cm that the end effecto of the vehicle must tack. Find the angula velocities of the wheels, and L, when the tacking speed is v cm/s 9. cm/s. Solution -a). Let C be the distance of the instantaneous cente of otation (IC) on the axle line, as shown in Figue S-. he atio of ight and left wheel velocities detemines the location of the instantaneous cente of otation: L C b () C b whee b cm. Fo the fist segment of the cicula tajecto C cm. heefoe, L. he tacking speed and the wheel velocities ae elated as: L v () v whee cm. cm and v 6.8 cm/s. heefoe, ad/s. It taveled fom point O to point B: OA v t. he time equied is t 5 sec. Similal, fo the second segment, the IC is at 6 cm. Fom (), L C b 6 C b 6 Using this in () ields L v v. ad/sec, L.67 ad/sec 6 he taveling time fo the second segment is: t 5 sec. he total taveling time is sec. See the plot below. Y L C C L Ov X 5 5 ime, sec Figue S- Figue S- L
3 -b). An motion of the vehicle is epesented as a otation about an instantaneous cente of otation, which exists onl on the axle line (including an infinite point). heefoe, point F must be on the axle line. As shown in Figue S-, the distance of the cente of otation fom the vehicle centeline, C, satisfies the following elationship: C ( EOv ) ( EF) C cm () Fom (), L C b, 5 L C b 5 () Let be the angula velocit of the vehicle otation. It is detemined b the instantaneous wheel velocities: ( L ) L b b (5) Since the tangential velocit of the end point on the cicula tajecto can be witten as v FE, (6) we can find v 5 5 Substituting this in (5) ields, b.6 ad/sec, 5 ad/sec L 5 L (7) Note that the fomula of () is valid fo point O v, and not fo end point E. Y ajecto of end effecto F: Cente of otation C 5 cm E v Ov cm End effecto O cm Figue S- X
4 Poblem Figue 5 shows a thee degee-of-feedom obotic wist at a home position. A Catesian coodinate fame, O x z, is placed at the base with the z axis pointing upwads. he length between oigin O and point W is, and the distance between W and endpoint E is. he fist joint otates Link about the z axis, and the second joint otates Link about the x axis in the ight hand sense. he thid joint is also a evolute joint, otating Link about the z axis. Coodinate fame O x z is attached to Link, O x z to Link, and O x z to the end effecte, i.e. Link. Figue 6 shows the wist configuation at /6, /6, and /. Unit vectos, ni, ti, b i, i,, ae attached to each coodinate fame pointing in the diections of axes xi, i, z i, espectivel. Answe the following questions. x z Endpoint E E b z t Link z z b x n x z Link W x Link n x x z b t t z n Link x O z x Figue 5 Figue 6 a). Find the diections of unit vectos n, t viewed fom the base fame O x z b computing thei components with espect to the base fame. Obtain the otation matix associated with the coodinate tansfomation fom O x z to O x z.
5 b). Find the diections of unit vectos t, b viewed fom the O x z fame. hen find the same unit vectos t, b viewed fom the base fame O x z. Compute the components with espect to the base fame. c). Obtain the coodinates of endpoint E viewed fom the base fame O x z. d). Is thee an othe combination of joint angles that oients the end-effecto exactl in the same diection? (In othe wods, ae thee multiple solutions to the invese kinematic poblem?) If multiple solutions exist, find the joint angles that oient the end-effecto in the same diection as that of /6, /6, and /. Also sketch the wist oientation of the multiple solutions. Solution -a). he z components of these two vectos ae zeo. On the x plane, these vectos point in the diection of /6and /, espectivel, as shown in Figue S - : t 6 6 n, t x Figue S - Concatenating these vectos togethe with b we can obtain the otation matix associated with the coodinate tansfomation fom O x z to O x z. n c s z ( ) n t b s c -b). he x components of these vectos ae zeo. Within the z plane the point in the diection of /6and /, espectivel. heefoe, t, b, x ( ) c s s c 5
6 Multipling z ( ) fom left, c s c sc ss n t b z( ) x( ) s c c s s cc cs s c s c t, b -c). Adding two vectos OW and WE ields OE b b xe ss e cc ze c -d). hee ae two solutions. See Figue S 5. Solution : /6, /6, and / 7 Solution :,, and ' +8 o +8 o Solution Solution Figue S - 5 6
7 Poblem Shown below is a thee degee-of-feedom plana manipulato am with link lengths,, and. he fist joint is actuated with a moto fixed to the base, while the second joint is diven b anothe moto placed at the base though a fiction-less belt-pulle mechanism. he thid moto diving Link is fixed to the tip of Link. Let,, and be, espectivel, the toques of the thee motos, and,, and be joint angles, as defined in the figue. Ignoing fiction and gavit, answe the following questions. a). Obtain the coodinates of the endpoint E, as well as the oientation of the end effecte, p x e e e, as functions of joint angles,, and. b). Obtain the Jacobian matix elating the endpoint velocit x velocities: q. p to joint e e e c). Figue 7 shows the am configuation of / and. Is this configuation singula? Justif ou answe mathematicall. Fo the est of the questions, use the following paamete values: m,.8 m,. m. d). Figue 8 shows the configuation of /, and /. At this configuation linea foce Fx and F N as well as moment Nz Nm act on the end effecte. Obtain joint toques to bea this load. Ignoe fiction and gavit. e). Based on the esult of pat d), discuss whethe the end effecte can be moved at a velocit aligned with the diection of the endpoint foce and moment, e.g. p m / s ad / s. Endpoint Link Link Link O x O x Figue 7 Figue 8 7
8 Solution -a). -b). -c). -d). -e). x c c c e e e s s s () s, s s, s J c, c c, c (),, Fo / and,,, J,,,, () det J ( ) () heefoe, the configuation is not singula. Fo /, and /,,,,.8, J,,,.,. (5),,,, Using the dualit law, Fx J ( F ).8. (6) N. he above esult shows that no actuato toques ae needed fo beaing a non-zeo endpoint foce. he load is bone b the stuctue. his is possible onl when the am is at a singula configuation, and the load acts in the diection in which the endpoint cannot move with a non-zeo velocit. heefoe, the equied velocit p m / s ad / s, which is in the constained diection, cannot be geneated. 8
9 Poblem Figue 9 shows a wheeled obot with wheel adius = cm and bod width b = cm. a). Obtain the coodinates of the instantaneous cente of otation, X and Y, when the vehicle is at the oigin O and the two wheel velocities ae = ad/s and L = ad/s. b). Figue shows the time pofiles of the two wheel velocities. Obtain the vehicle location at time t f = 6.8 sec = sec, X ( t f ), Y( t f ), ( t f ), given that the vehicle was at the oigin at t =. c). he toque constant and amatue esistance of the two motos ae Kt and, espectivel, and the gea atio of the gea head is a: (a >). Obtain the electic powe P consumed in both motos when the vehicle is staing on a slope of /6, as shown in Figue. Assume no slip between the tie and the slope and no fiction at the geaing and othe mechanical components. he vehicle weighs m kg. Y b X Figue 9 Wheeled obot. ad/s time sec o mg Figue ime pofiles of wheel velocities Figue he vehicle is staing on a slope Solution 9
10 -a). he instantaneous cente of otation exists on the line connecting he two wheels, so that the velocit vectos of the wheels ae pependicula to the line, AC. See the figue attached. Since the two tiangles ae simila, l BC BC AC () BC b Substituting the given values to () and solving it fo BC ield BC cm () heefoe, X, Y cm () B A C Anothe fomula to use is the cuvatue - wheel velocit elation: b( L) cm L -b). Note that the velocit time pofiles ae piecewise constant. his facilitates the integation of the velocit elationship given b: X cos cos Y sin sin () L b b A gaphical appoach Fo the fist. sec, the vehicle otates aound the cente of otation b ( L) t ( 5) (5) b he adius of otation is b L b cm (6) L Fo the second. sec. ( L) t ( 5) (7) b b b 5cm (8) L Fom the diagam above, the coodinates of the vehicle at t = 6.8 sec ae: X ( t ) 8 cm, Y( t ) 8 cm, ( t ) (9) f f f
11 F 5 cm Y C Cente of otation fo the fist. sec. cm E Cente of otation fo the second. sec. D O X he standad numeical appoach -c). Fom the thid ow of (), ( t) ( L) t t () b Using this fo integating Xt () ove t., X (.) X ( )cos ( ) L t dt () X ( L) cos ( ) L t dt cm b Similal, Y(.) Y ( )sin ( ) L t dt cm () Fo the second half, ( t) ( L) ( t ), t b epeating the same pocess and combining the two peiods ield the same esults as the gaphical appoach.
12 o sta on the slope (with zeo acceleation), the actuatos must geneate a total taction foce of mg/ N, whee g is acceleation of gavit. his equies each actuato to geneate mg Nm () about the wheel axis. Powe loss at each actuato is given b Ploss i () Ka t heefoe, the total powe loss at both actuatos exeting the equied toque is given b P otalloss m g (5) K a 8K a t t
13 Poblem 5 hee ae man inteesting featued to exploe in biped locomotion. Figue shows a simple one leg model whee the foot is fixed to the floo. wo DC motos ae attached to the ankle and the knee joints. he ankle moto geneates toque between the foot and Link, wheeas the knee moto geneates toque between Links and. he length of both links is, and joint angles,, ae used, as shown in the figue. a) Obtain the Jacobian matix elating the hip velocit, xh, H, to joint velocities,,. b) Obtain the joint toques,,, to bea the gavit load Mg at the hip. c) Obtain the powe consumed at both motos fo beaing the Figue Single leg model gavit load in Pat b), and find the leg configuation whee the powe consumption becomes the smallest. Assume that both motos have the same popeties: toque constant Kt and amatue esistance. Futhemoe, both motos have the same gea educe of gea atio :, with >. Figue shows a tajecto of both legs. he foot of the pivotal leg (Leg ) is fixed to the floo at Point B, while the othe leg (Leg ) is moved fowad fom Points A to C along the tajecto shown. Note that the knee of Leg is full stetched thoughout the pocess,. Figue -a shows a simplified kinematic model whee the two links of Leg ae teated as a single link since. A new set of joint angles,,,, ae defined as shown in the figue. Fig. b shows the leg Figue ajectoies of both legs configuation whee /, / 6, /. d) he tip of Leg, Point C, is moved b each of the thee joints,,. Show gaphicall the velocit of the tip of Leg geneated b each of the thee joints denoted b J, J, J, espectivel. Show cleal in which diection each vecto is pointing in ou diagam.
14 e) V H At the leg configuation in Figue-b, we want to move the hip hoizontall at a desied speed, wheeas the tip of Leg V at in the hoizontal (x) diection. Find the joint velocities,,, to meet these equiements. (a) (b) Figue Simplified model of both legs Solution -a). he coodinates of the Hip ae given b x H H cos cos( ) sin sin( ) () Diffeentiating these ields the Jacobian x H H J, whee s s s c c c J () -b). Fom the dualit of kinematics and statics, F x J F whee the endpoint foce is F, F Mg. x s s c c c c Mg s c Mg c () -c). he powe dissipation in one moto is given b
15 P loss i K () m t Fom Pat b) each moto toque equied fo beaing the gavit load is given b Mg Mg m ( c c ), m c (5) he total powe dissipation is then Mg [( ) ] t Ploss c c c K (6) which is non-negative. heefoe, the minimum of the total powe dissipation is ; this occus onl when c and c (7) c which implies cos. heefoe, and, (8) Figue S- he leg configuation of zeo powe consumption he onl meaningful configuation is,, which is shown in Figue S-. he leg is at a singula configuation, and the gavit load is bone entiel b the stuctue. Note: A moe igoous method fo finding the configuation of minimum powe consumption is to take patial deivatives of Ploss (, ) with espect to, and set them to zeo. he same conditions as (8) can be obtained fom them. -d). Let ( x, ) be the velocit vecto of the tip of Leg. he Jacobian matix elating ( x, ) to joint velocities,, contains the thee vectos J, J, J : x J J J J J J (9) Note that vecto J is the patial deivatives of x, with espect to, while the othe two joints ae fixed. Namel, both legs can be teated as a single igid bod. As the fist joint alone otates, the tip moves Figue S- he velocit of the tip of Leg geneated b each of the thee joints, J, J, J.
16 along a cicle of adius O as shown in Figue S-. heefoe, vecto J is pependicula to line O. Similal, J is pependicula to line H, and J is pependicula to line K, as shown in the figue. Note: No need to compute J, J, J. Moe impotantl, gaphicall show the diections coectl. Each of J, J, J must be pependicula to a specific line, as descibed above. -e). he hip velocit is detemined b the angula velocit of the fist joint alone: VH. heefoe, V H () At the given configuation, column vectos J, J ae aligned with the x axis, wheeas J alone has some component. See Figue S-. Fom the figue, the components of the thee column vectos can be obtained as ( ) J,, J J () Fom (9), the tip velocit is given b x ( ) J J J () Compaing this to the desied tip velocit: x V () we can immediatel find, fom the components. Fom the x components we can also find: V ( ) () Substituting () into the above ield V VH (5) 6 Answe: V H, V VH, 6 (6) Figue S- he thee column vectos at the given leg configuation
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