12O Chapter j' Dfferential Motions and Velocities

speedsofeachandeveryjoint'atanyinstant,suchthatthetotalmotioncausedbyrhe robot will f..q*l to ih". d.rir.d sp!"d of the frame. In this section, we will first study

the difterenrial motions of a frame. Then, we will study the differential motions of a robot

mechanism' Finally, we will relate the two together'

3.6 Differential Motions of a Frame

Differentialmotionsofaframecanbedividedintothefollowing:

o Differential translationso Differential rotations

a. . pin-.r."it1 transformations (combinations of translations and rotations)

3.6. 1 Differential TranslationsA differential translation is the translation of a frame at dift-erential values' Therefore' it

canU. ,.prJ.ni raiy m*r1dx, dy, dz).'T11is means the frame has moved a differential

amount along the x-, !-, and z-axes'

Example 3.2A frame B has rranslated a differential amoqnt of Trans(\'01,0.05, 0.03) units' Find

its new location and orientation'

io.zoz o -0.707 slI o 1 o 4l

B:l^-^- ^ I

| 0.707 0 0.707 e I

I o o o 1.]

Solution: Since the differential motion is only a translation,.the orientation of the

frameshouldnotbeaft.ected.Thenewlocationoftheframeis:.707 0 -0.707010-707 0 0.707

00,0

3.6.2 Differential Rotations about the Reference Axes

A dift-erential rotation is a small rotation of the frame' It is gen-erally.represented by'irt(iiafil,*nlar

rrr."r, that the frame has rotated an angle of d?,,abo"tr an axis 4'

Specifically, differential rotations about the x-, y-, and z-ax.es are definedby 6r , .3y, end 62.

Since the ,or"rlorrr ".. small amounts, we can use the following approximations:

sin6x: 6x (inradians)

cosdx: 1

5.01-l

4.0s I

e.03 I1l

I

5l f

0

4l_ |nl- lo1l L

o.o1l lo.toto.osl I o

o.o3 lx lo.zoz1i I o

i1

": [:lo

00100100

0 -0.707100 0.707

00

3.6 Dffirential Motions of a Frame121

Then, the rotwi1 be:

:atron matrices representing differential rotations about the r- , y-, and z_axes

ft o o ol f r n ^., n-rt',{*,a*):f!

u,- -,,,,

Il,^,,,,,u,): I ; ?'J Sl D r- ir -da 0 0l

lo; ;;i IlSi?J'*'e'*r:1'; t;ilsimilarly, we can arso define dift-erential rotations about the current axes as:

Q'11)

f,ll o -or, Sl l1 ? so ol fr -6a o orao4n'an):1f, ;, ;" Jf,o'1''o,l:l-ou, i I If,^,,r,u,,:/i

-:'iSllooorJ l;';;il Lo,oo,l

Notice that these matrices defi, rha -,r^,,.^ L Q'12)

of unit vectors. a.. :.-::otrythe;&:ve!aiestablishedpreviouslyaboutthemagnirudedifferentialva".rr"*,.fll]:'l^t1'+ (a)l ) 1'.However, as you -., ,.-.;;;;;l,.i:id.,:g;.-Gr,#:?:'"",:T,;#ri:.T:,5il*.:y*H#1.#trf; ::1*;;drtterentials such as (6x)", the magnitude if ,h.,r..tor;;;#;.:eptabre.As we have alreadv.seen, if the"o.a..lr-,rrripri.*i"".i*"rji.r.*r"nges,

rhe resulrwill change as wen. fir._l.r".., i; -"*" -i*i."ri;"lr.J".i'-ng the order of mar.ffi l.,.;;i-'#ffi :th',:ff **il,;#ffiffi";TffJilf; '**..,,torders,we

Rot(x,6x)Rot(y,6y) :

Rot(y,6y)Rot(x, tx) :

I[;if,f}' ;lilt*;?ilfj,;'!il ff ;j il:l+r{ ?ll

However' ifas before' we set higher-order differentiars such as 6xty to zero,the resurts areexacrly the same. consequendyl r". aia-...rri"r *"norrr, ,i. oi#ofmurtiplicarion is noronger important ^o f:k:;)ilQ;i;'i 1,,,r,;;1;;;(;,;;j: rhe same is true rorother combinations of ro_tarions, ln.t,rang amrJ,r,rii,-;;il;r;"rt the e_axis.You may remember r.o- y."ur-;;;"t?ri...."urse that-large-angle rorarions abourdift'erent axes are not iommutative, and therefore, cannot b. a-dd.a in different orders.For example' as we

1?:: {:*it r.."].ar"" rorare an object 90" about the r_axis,followed by a 90" rota1L1 abo.rr it . r-orr, irr. ,o.rrr-.n iu #.,;e;., rr you reverse rheorder. However. velocities are .o__or"rirre.and c.an U.-"aa.Jl, vectors; therefore,9 : ,"i * arj -f ot.k, regardless of ,t. ora.r. ffri, ir' ,;;;""re, as rhe abovedrscussion shows, ifwe neglect trr. rrigi.._ola.. aia_.r.rrrd;;;.;;er ofmultiplication

122

Chapter 3. Dffirenilal Moilons and Velocities

Figure 3.5 Differe n

wirr be unimportanr ,::::::""- about a generai axis 4.

same is r*. h;;i: rrnce velociries are in faadifferential

modons divided by time, the

Example g.g

#"jfijTilffYiJ;l"jlsiT#rT,1 bv smarl rorations abour the

Rot(q,d0) : po71*,dx)Rot(y,6y)Rot(2,u.) :[ +

- . !{.l-5l 3x 1[o o o

ololo / Q.14

1J

3.6.3 Differential Rotation aboutBased on rhe above, since rhe o.o

t a^"t"ral Axis q

ji:;1l##i{i#liffi iff**,r,':'dlr:il:.?,::ffi ;q'fr ,;::r:,il"J..,,.n,.",ir:::;,4ru1*:gfi tl',';'J,o;'if,:l:h.,.".o'ni.,n,i.'"11tiorx,

Ro r (q, d 0) : f,t

G, !4

a", (0, o *lit,,o" *'"v gl" :,:i'il: r:?t"t:'#:*t;, Ill A -,,._r,_,"r1<,oZ)l'u o oll

/o r -dx ;// 1' s 5v

/: ,j ;" :ll -',, j iLo o e t]lo e o

tocities

re, the

is notNSUme

tationss).,d as:

(3.1,4)

t23j.6 Dffirential Motions of a hame

solution: Substitucing the given rorarions in Equation (3.,14),,we.get:

0.05--0.1,1

0

Note that the lengths of the three directional unit vectors are 1.001, 1.005, and1.006, respectively. If we assume that 0.1 radians (about 5.7") is small (differential),these values are acceptably close to 1. Otherwise, we should use smaller values fordifferential angles.

where

I r -62 6y ol I r _0.02

Rot(a,dil:l\ 1 -6x ol 1o'oz 1-",-l-u, 6x 1. ol:l-o.os G1

lo o o 1J I o o

ol

il

3.6.4 Differential Transformations of a FrameThe dift-erential transformation of a frame is a combination of differential translations androtalions in any order. If we denote the original frame as Tand assume that dT k thechange in the frame 7 as a result of a differential transformation. then:

[f + dT] : lTrans(dx, dy, dz) Rot(q, dl)ltTlor [af1 : lTrans(dx, dy, dz) Rot(q, d0) _ 4[f]

where lis a unit matrix. Equation (3.15) can be written as:

(3.1s)

Idrl : taltrlf\l : [Trans(dxo dy, dz) x Rot(q, de) - 4

(3.15)

[A] (or simply A) is called dfferential operator.It is the product of dift-erential rranslationsand rotations, minus a unit matrix. Multiplying a frame by the difierential operator [A]will yield the change in the frame. The differential operator can be found by multiplyinjthe matrices and subtracring the unit matrix, as foltws:

A, : Trans(dx, dy, dz) x Rot(q, d0) - I

(3.17j

A-

| -62 6y ol [r 0" o ol3z 1, -6x ol lo 1 o ol-6y 6x 1 ol.-lo o I olo o 0 1l Lo o 0 ]j

[t o o a*11

l; ; ? '/.lllo o o 1JL

f o -62 6y

|6= 0 -drl-rr 6x o

L0 0 0

d*1dyl,;)

As you can see, the differential operator is not a transformation matrix, or a frame. Itdoes not follow the required format lither; it rs only an operator, and it yields the changesin a frame.

124Chapter j' Dffirential Motions and Velocities

ExamPle 3.4Writethedifferentialoperatormatrixforthefollowingdifferentialtransformations:dx: 0.05, dy:0'03,"il:_.o'otunitsanddx : O'02' 6I: 0'04' 6z: 0'06radians'

Solution: Substituting the given values into Equatio n (3'77)'we get:

ExamPle 3.5Findtheeffectofadift.erentialrotationof0.lradaboutthey-axisfollowedbyadifferential translation of [0.1, 0, 0.2] on the given frame B.

Solution:Aswesawbefore,thechangeintheframecanbefoundbypre--"1tipl-v'1qthe frame with the d;tr;;;rtrl op.ri'ror. Substituting the given information and

multiplying the matrices, we will get:

dz :0.2, 6x : 0, 6Y : 0'1, 6z:0

I o -0.06 o'04 o'os lI o.oo o -o'02 o'03

I

a : I -0.04 o.o2 o o'ot

I

L o o o ol t

For dx - 0.1'

ldBl: [Al[n1 :

[o o 1 tolIt o o s lu:lo 1o 3l[o o o 1l

o o.r o.1l[o o I lol l-9 o:t

s s +lli ;l il:Ls s

)",-Au I - "t

fol0| -0.1L0

0

0

-0.10

0.4 I0l=0.s

I0lI

3.7 Interpretati"tt "f th" Diff"t"ttti"l C

ThematrixdTinEquations(3.15)and(3-16)representsthechangesinaframeasaresultoiAf.r.n lal motions' The elements of this matrix are:

I dn- do* da, OO"f

I On, do, do, dP, I

I on, do, da, dP, I

Lo o o ol(3.18)dT:

d Velocities

formations:06radians.

llowed by a

-multiplyingcmation and

(3.1s)

o o.4J

l'-l'lI

1-'8 Dffirential Changes between Frames r25

Tn*: dT I Tal (3,1e)

Exarnple 3.6Find the location and the orientation of frame B ofExample 3.5 after the move.Solution: The new location and orientation ofthe frame can be found by adding thechanges to the original values- The result is:

Br*: Borisinat * ilB

The dBrnatrix inExample 3.5 represents the change in the frame B, as shown in Equation(3'18)' Therefore, each element ofth. matrix represents the change in the coresp;"di";element of the frame. As an example, this -.im that the frarrie moved a differentialamount of 0.4 units along the x-axis, zero along the y-axis, and a differential amount of-0'8 along the e-axis. It also rotated such that tiere was no change in its n-vector, therewas a change ofo.1 in the o, component ofthe o-vector, and, ot".rg. of -0.1 inthe a=component of the a-vector.

The new location and orientation of the frame after the dift-erential motions can befound by adding the change ro the frame:

ll.ll-fl= lolo

01001000

0.1 0

000 -0.100

'))

-;)+l l;

0.1

0

t

0

7

0

-0.10

IDifferential Changes between Frames

The difterential operator A in Equation (3.16).represents a diSerenrial operator relacive tothe fixed reference frame and is technically uA. ilowever, it'is possible to d.fi"; -;;;;diffe':ntial operator, this time relative to the currenr frame itsil4 that will enable us rocalculate the same changes in the frame. Since the differenrial operaror t.il;.;; ri;frame (7A) is relative to^-"..rr..rrt !ame, to find the changes in th! frame we musr posr-multipiy the frame by TA

1as we did in chapter 2).Ther.r",rlt *iil b.;;#;,il;il;;operations represent the same changes in the frame. Then:

[dr]: [a][7] : [rl'a]'--+ ir-1] [a][r] : lr-'l lrll'al-+ ['a] : lr-1l1n1Jr1

The^refore,Eg11t-ron (3.20) canbe used to calculate tn. ar"..rrtaal operator relative tothe frame, I A. 'We

can multiply the matrices in Equation Q.2,0) andsimplify the resul as

(3.20)

126

tbllows. Assuming

t-t"T-': lo'

lo,L0

rdx:n.[6xp+d]'dy:o.[6xprd]rd.:a.[6xp+d]

See Paull for the derivation ofthe above equattons.

Chapter 3. Diferential Motions and Velocities

that the frame T is represented by an

fot^A: I u:

| -aYL0

I o -26. ,6y r dxf[T-1][A]tTl :7A: |'9:l-'sy rl,x o rdzl

I o o o oJ

(3.21)

As you can see, the rA is made to look exactly like the A marix, but all elements arerelative to the current frame, where these elements are found from the above multipli-cation of matrices, and are summarized as follorvs:

'dx: 6.n167-6.ordz: b.a

(z'>ct

Example g.ZFind 8A for Example 3.5.SolFtion: 'we

have the following vectors from the given information. -we willsubstitute these values into Equatii n (3.22) and will calculare ,r..,o.r"bd ;;; ;6.

n:[0,1,0] o:[0,0,1] a:[1,0,0J p:[10,5,3]6: [0,0.1,0] 6: [0.1,0,0.2]

6xp- [0.3,0, -1]

thus6 x p * d: [0.3,0, -1J+ [0.1,0, 0.2]: [0.4,0, _0.8]Bdx :n' [6 x p + d] : 0(0.4) + 1(0) + 0(_0.8) : 0udy :o. [6 x p +d] : 0(0.a) +0(0) + 1(-0.s) : _0.8Bdz :a. [6 x p+d] : 1(0.4) +0(0) *g(_o.s) : 6.4Bdx: b.n : o(o) + 0.1(1) * o(o) : g.1

u6y :6' o : 0(0)+ 0.1(0) +0(1) : 0 .:Gi,

86z :6. a : o(1) +0.1(o) * o(o) : 9

n, o, a, p matrix,

-32 6y d*f0 -dx dvl6x 0 drlo o o_l

we get:

ny nz -p "loy oz -p.ol ,I anoay az -p'aloo 1j

0

10

Q,2I)

3.9 Dffirential Motions of a Robot and its Hand Frame 127

Substituting into Equarion (3.21) yields:

Bd : [0,-0.8,0.4] and n5 : [0.1,0,0]

As you see, these values for BA are not the same as A. However, post-multiplying the B.matrix by uA will yield rhe same result dB as before. -

Example 3.8calculate BA of Example 3.7 directly from the differential operator.

Solution: Using Equation (3.20), we can calculate the BA directly as:

[o o o ol"a : I3 ol, -0 1 ;"'I

L0 o o o_l

10.l

sl3l1Jll, I l ;l l;lual : [B-1][a][B] :

10 -501-300-1000 1

00

foloIrLo

f0l0loLo

01001000

0

-0.80 -0.10.1 0

000.4

0

which, of coune, is the same result as in Example 3.7.

3.9 Differential Motions of a Robot and its Hand Frame

it

In the previous section, we saw the changes made to a frame as a result of dift-erentialmotions. This only relates to the frame changes, but not how they were accomplished. Inthis section, we will relate the changes to the mechanism, in ihis case the robot, thataccomplishes the differential motions. We will leam how the robot's movements aretranslated into the frame changes at the hand.

The frame we discussed previously may be any frame, including the hand frame of arobot. dTdescribes the changes in the compon.nir of the n, o, u, f lr..rors. If the framewere the hand frame of the robot, we would need to find oui how the differentialmotions of the joints of the robot relate to dift-erenrial morions of the hand frame, andspecifically, to dT. Of course, this relationship is a funcfion of the robot's configurationand design, but also a function ofits instantaneous location and orientation. For elxample,the simple revolute robot and the Stanford Arm of Chapter 2 would .equire -rlerydifferentjoint velocities for similar hand velocities since their configurations are different.

r28 Chapter 3. Dffirential Motions and Velocities

However, for either robot, whether the arm is completely extended or not, andwhether it is pointed upward or downward, we would rr..d rr.ry di{l-erentjoint velocitiesin order to achieve the same hand velocity. Of course, as we discuss"ed before, theJacobian of the robot will create this link berr,veen the joint movements and the handmovement as:

dx

dy

dz

6x

6y

6z

d0t

doz

doz

do+

dos

doo

RobotJacobian

or IDI -- ullDrl (3.10)

in [A], the differentialSince the elements of matrix [D] are the same informatron asmotions of the frame and the fobot are related to each other.

3. 10 Calculation of the JacobianEach element in the Jacobian is the derivative of a corresponding kinematic equationwith respect to one ofthe variables. Referring to Equatio" 1a.lo;, tf,e first ele-erriln Jo1is /x. This means that the first kinematic equarion L.rrt ,.pr.r.rrt -orr.-.rrt, "lo.rg

ih.x-axis, which, of course, would b, pr. In other words, px expresses the motion ofthe hand frame along the r-axis; consequently, its derivativ. -itt U. dx. Thesame will betrue for dy and dz. considering the n, o, u, p matrix, we may pick the correspondingelements of p*,.yt, and p. and differentiate them to get dx, dy,'and, dz.

. A1 an example, consider the simple revolute ar- ofE*"mple Z.ZS. The last column ofthe forward kinematic equation of the robot is:

lr,]:l?'triiiil:ii,f7r](3.23)

Taking the derivative of p* will yield:

P*: Ct(Cr.ooo * C4a3 * C2a2)

, 0p- .^ 0p- .^ 0o.,apr: ffide, +

ffid?z * . .. +ffide6

dp,: -St[Cruooo I C2a3 + C2e2]dq * C{-Syaa4 - Sl;.a3 - S2a2]d02

* C 1[- Syaaa - Sya3]d?s I C 1[- Syaaa]d0 a

ocities

ru can

,il"tly,: three'd9y as

ions, ifconffolrol the

ry have

lare any

ints?

a

I

,

i ..

Posluonpu mayie, these

lrotions

I

oftheareGr.

jointa robot.

ove.1n

uauonsrobot in

ofle of

r43hoblems

1. Paul, Richard P', "Robot Manipulators' Math-

ematics, Programming, and Control," The MiTPress,1981.

IIr. Koren. Yoram, "Robotics for Engineers"'

McGraw-Hill, 1985.

Fu, K. S., R' C. GonzaTez, C' S' G' Lee'

"Robotics: Control' Sensing, Vision, andIV.

Intelligence," McGraw-Hill, 1987'

V. Asada, Haruhiko, J J' E Slotine, "Robot

Analysis and Control," John 'Wiley and Sons'

NY, 1986.

VI. Sciavicco, Lorenzo, B' Siciliano, "Modeling

and Control of Robot Manipulators"'

McGraw-Hill, NY. 1996'

3.1. Suppose the location and orientation of a hand frame is expressed by the following matrix' what is the

effectofadifferentialrotationof0.15radiansaboutthez-axis,followedbyadifferentialtranslarionof

[0.1, 0.1, 0.3]? Find the new location of the hand'

p;rH-

t-00121lr?Billo o o 1l

3.2. As a result of applying a set of differential motions to frame Tshown' it has changed an amount dTas

shown. Find the *;g"ir"a. of the differential changes made (dx, .dy,

dz, 6x, 6y, 6z) and the

di{ferential operator with respect to frame T'

0 -0.1 -o."i[ 0.6 I0.1 0 0 .0.5

|

-0.1 0 0 -0.s1,0 0 0 0l

ft o o sllo o 1 3l ).r-T:1" -1 n sl dt:

"l

lo o o rl

-33.Supposethefollowingframewassubjectedtoadifferentialtranslationofd':|700.5]unitsandadifferential rotation of 3 : [0 0'1 0]'

(a) 'What is the differential operator relative to the reference frame?

(b) What is the differential operator relativ€ to the &ame '4?

A-[o o 1 lol

I'rsll

144 Chapter 3. Dffirential Motions and Velocities

3.4. The inicial location and orientation of a robot's hand are given by T1, and its new location andorientarion after a change are given by 72.

(a) Find a transformation matrix Q that will accomplish this transform (in the (Jniverse frame).

@) Assuming the change is small, find a differential operator A that will do rhe same.

(c) By inspection, find a di{ferential translation and a differential rotation that constitute this operator.

For the given differenc:ion, and corresponding I

0

0.1

-0.10.20.20

3.6. Two consecutive frames describe the old (71) and new (72) positions and orientations of the end.-j:3-DOF robot. The correspondingJacobian relative to !, relatingto rtdz, Tr6x, Tr3z, is also gir-;-_Find values ofjoint differential motions ds1, d02, dfu of the robot that caused the qiven frame char+

3.7. Two consecutive frames describe the old (71) and new (72) positions and orientations of the end r:3-DOF robot. The correspondingJacobian, relating to dz, 3x,62, is also given. Find values of-ierlrdilferential motions ds1, d02, dfu of the robot that caused the given frame change.

3.8. A camera is attached to the hand frarne T of a robot as given. The corresponding inverseJacob:'- r-the robot at this location is also given. The robot makes a differential motion, as a result ofwhic': :change in rhe frame dT is recorded as given.

Find the new location of the camera after the di{ferential motion.

Find the differential .operator.

Find the joint differential motion values associated with this move.

lr o o 5l I t o 0.1 4.81

r:19 o --r 3l r,:lor o -r ,rl't-lo r o 6l '2-lo I o 6.21

L0 0 0 1J L0 0 0 1J

3.5. The hand frame of a robot and the corresponding Jacobian are given.changes of the joints, compute the change in the hand frame, its new locat

[o 1 o 1ot fj. 3 ? 3 3 S]

t:119 t, :l r^r:l9 'l 9 9 9 9l'"-lo o -t ol lo i o o l ollo o o 1l lj, 3 3 ; 3 ?l

De:

'' : f ; : l i] '': 14,

-";'' -t,, ?'] ": l: ? il

'':[; :ll] ':ll;,' :"1"t,1':i: ? ll

(a)

(b)

(c)

(d) Find how much the dift^erential motic 745

measuredrerative,.{i.";-i;;;;"""il13..iilif.ffi :i?f i;1TijT;T;i:;*r,"

,:f; iq:i l;3_iiiil t-o^*q-01 oTsj':13s;'J "-':/i -';'3 3?s/ dr:l e oo3 ; o^o?r

' l? Stistl 13-01 31-J3'9' A camera is attachedto the hand ftame T of arobot as given. The co:the robot relative to the a"-.

", ,rrtrl".*;; * also given. rrr".ouor llsponding

inverseJacbbian of

;H', ;:'""-'Ti::::Jili:".,Tffi*H*:;":-" a differen'lia'1 rir;;;;;' ;

(b) Find the differential operator.(c) Find the joint differential motion values D6 associated with this move.

fqlo:r l: I oooor': /i i -i tl ., ,:l:,

-',' r' s i ;/ro o o rJ l? ; s ; I l/

l-10. The hand fiame 7,, of o.^r-^. :- ,:ro cati on *r".,3niJ,l#l :: ;, f [:_TH:ffir":HH,

- frame described as;

Problems

. (b) Find the change in the {iame.

| -o.oz o _0.1 o.t lar:l o o.o2 o o.oslI 9 _0.1 o _o.s

IL o o o oJmv:rse Jacobian of the robot at thisa differential modon relative to this

rHD : 10.05 0 -0.1 0 0.1 0.1J..{") fiig which joints must make a differential rindicated differential motions. nofion' and by how much, in order to create the

\_/ r 4r,s urL lrrarrge tn the frame.(c) Find the new locarion of the frame after the differentiar motion.(d) Find how much the differential rnodons (given above) should have been, ifmeasured relative to

the lJniverse, to move the robot ,. ifr" ,._. new location as in pan (c).

fele3t li 3_"33s1*:l; 3 _', :l ."r-,:19 -'02 o o o ol[o o o ;] l; ;, 3 ? J SlL1 o o o o ill- The hand frame Zofa robot is given. The correralsoshown. Tt

"rouot*"t .sadifferentialmodo.tlordl]rginveneJacobian ofthe robot at this ]ocation is,,**T.j#il,ff# ;,,,ffi .:::T

::: ;'::: j..,,1 l_: J: j,]:(b) Find the change in the frame.

146

(c) Find the new locrti^- ^r,L- . , u'tupter J' Difr

(d) Find n"* ,',.loiillTij.: frame afre* ntK ::::-"t Motions and v

(d) Find how much th^ A:rr.^,- -.

:-"" drrer the dtflerential modon.

Fram. a r" Tli3:*,ff::::TJons (given "uo,..),r'ouia have.b.een, irmeasured ierativesame new location as in part (c).

'fo 1 o 3l f: o e oool.: l; 3 -i ;/ ,-':l: -3: ;' 3 s sl

ro 0 0 ;l /i ;' 3 I J siIro3.l2.Ca1culatethe\J Lr u o o o ll3,13. 6"1.ur"te the rrrzr ;lement

of theJacobian for the revolure rohot nrn_.^..,_ rJ'lJ. Calculate the ToJ.. el._^_. ^".:

- :*"":'"u

ror the revolute rob3.14. Using.nrrrrrorr'tu

element of theJacobian for the..r.orurJ;;;::;:ffffir:;r?.q ua d on s ro, r.,J,rol*.*,ffi :nff .^.il.T *,: *

". ;;;,.tt. t,.,, rn*""',r.6) .r;r+;*^--, cvlindrical .*"rl"ffijit1"::f:":l,l -'f ."-

" - ,Y;;,?"::ff fi # ;?i.$fj**|*ii I "ffiT: T.: ",r,. ..,ir,p "il ":J ffflr.tu-.o.".r'"in.ri.uoot,thethreejor.,rr,.,o.,1,lfhericat;;;;#J;J:t:Tf"ffi::":tJ'#i;

components of the velociry * ;;;ilrff:.are given for a corresponding location. Find the tr'r:0.'l.in/se" i,-nnF

"' *;:*;ffi;#;u,'^: o' ^'n:, r : 15in, a : 30.,, :,Q'p

: velocity of the hand #.1* given for a corresponding location. Find the thre

r:2in/sec,.f : 0.05 rad./sec, i, = 0.1 rad/sec, r: )13'18' For a sphericai robot, the il;;r; ""::-,:,_=

o'1' nd/sec, v : 20 in, f : 60o, )/ : 30o.componenb "u

#:i,Tihreejoint veloc''re velociry .drh;;;;Hare glven for a corresponding location. Find the three

i: I unirl"o- p - ,

3.19. For a cyrindricar tnitlsec' l): l rad'/sec' i' : l'radlsec. r: 5 uni

;H:'T#;?..1,,0.11;it'H:'ff:."r"n':.',*f :i"..,H jj*{#.-.11",,".".."

"--^! vLruLraes $rat will generate ,fr. !""r, frrrrJ

3-20. For " ,0n"r..1..:1inlsec'

y:3in/sec'i:5inlsec' q:45o' ' =*'otn, l:25inL.

:

iilsnolain, ,*]',,*'l;:T: ::fi',.:Til:.::ii"",l"f:iy.nll" li:o riame are siven ror arrame verociry nree joint velociries rhat w'r *."..rr. *. ilJ* n*oi : 5 in,/sec. y : 9 in/sec, z :6inlsec, f :60o, y:20in,)/:30o.