tài liệu robot công nghiệp

ve ,o a6 .a.u t+o tdng qudt cria b0 didu khidn Robotlong c6ng nghi€p?. D6p 6n C6 2loai didu khidn Robor theo vdng kin: -- Didu khidn tlidm-didm: trong thdi gian dinh trudc di chuydn dd

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Trin! biy phrrdn g ph6p didu khidn Robot theo vbng tir ve ,o a6 a.u

t+o tdng qudt cria b0 didu khidn Robotlong c6ng nghi€p?

D6p 6n C6 2loai didu khidn Robor theo vdng kin:

Didu khidn tlidm-didm: trong thdi gian dinh trudc di chuydn ddu c6ng

t6c til vi tri hu6ng A sang vi rri hrrdng B, c6c tqa d0 trurdng atrd,;il;;;

Robot bXng bQ lQp trinh bhng tay (teach pendant)

- Didu khidn theo.qu! dao:'roa d0 ddu .ong t6c li hdm theo thdi gian,

qu! dqb ddu cong tdc thtrdng ri drrdng trring hay drrdng cong rrong kh6ng gian

viQc tinh todn didu khidn rhrrdng drrgc thrlc hien bdi c6c bQ vi xt tf hay

mdy tinh cdc girii thuQt didu khidn thudng ding li di6u khidn po, prn,

ngiy nay cdn c6 c6c giii rhuat didu khidn rh6ng minh kh6c nhrr: logic

md (fuzzy logic), m?ng neural (neural network),

Ca'u tg9 tifu qu6t cfia b0 didu khidn:

Ndu tay mdy c6 n kh6u, m5i khau c6 mQt vi didu khidn vdL mach c6ng

sua't didu khidn vf trf dQng cd, vi tri vd v6n tdc di chuydn dddc rinh to6n

bdi vi xfi ly ho4c m6y finh BQ Teach pendant fhr/dng i,ro" a,ing ad ai6u khidn Robor bxng tay vi cfrng c6 thd l6p rrinh cho ciic hoar dg,g ddn

gi6,n cfia Robot (didm-didm)

ROBOT

chuung 2: DQNG HQc TEUAN RoBoT conc NGETEP

l NQi dung bdi todn dQng hgc thuQn

2 Cdc ma trQn quay co b6n vima tr{n chuydn ttdi thudn nhdt

3 Quy tdc Denavit-Hartenberg (D-H)

4 Giei bai to6n dQng hqc thuQn Nsfrn hdne cfru hdi vd it(io tin chi fiA chrone 2

I CAU h6i Cho tay mr{y c6 2 khdp guaY, 1 khdp tinh tidn nhrrhinh v6

VE cdc hQ tryc l6n tay mdy theo quy t{c D-H.

Tinh ma trfn chuydn vi tt ddu c6ng t6c vd hQ toa tlQ gdc

Tim vi tri cria tldu c6ng t6c khi bidt 4 =10(cm), 02 = 0, 0, = n I 4.

5

D6p 6n a Cdc hQ trUc gdn l6n taY m6Y

b Tinh ma trQn chuydn v!

- Xdc dinh thdnp sdDH

- Ttnh cdc ma irQn chuydn vi thdnh phln (1d)

1

3

i

{

,-l

tp

* *'

ii, ili

'J,

h.

,i{, ,ir

i,

ill ll i t: i'

A lrcrl

0 lr", I

101

c

[t o o d,i fr, s2 o lrrrl

[o o o r.] [o o o r _]

- Til dd tinh ma trQn chuydn vi tit gdc ddn ddu c1ng td

Tro = Tro Trt .Tr'

lrr, - szr 0 d, +lrc, + lrcrrl

f o

- | ," czt 0 lrs, +lrsr,

I

I

Lo 0 0 I _l

c.finh vi tri ctia ddu c6ng t6c NhQn xit: vitri tIAu c6ng tdc so vdi h0 tqa clQ g6c chfr

f dr+lrcr+lrcrrf

Lol

Suy ra vi ( ni cria ddu c6ng Ji' t{c li:

I x = l0 +1, +aal,

]::f,,

I

2 Cdu h6i Cho tay mdy c6 Zkhdp 9u4y, I khdp tlnh tiSn; h0 tqa tlQ gdc &Igc dAt nhu

hinh v6

a YE cdc hQ rUc l6n tay m6y theo quy tdc D-H

b Tim ma trfln chuydn vi tr} tldu c6ng tdc vd gdc toa d0

c Tim vi tri cria ddu c6ng t6c khi bidt d2 = S(cm) , 0, = 0, 0, = n I !, .

a

5

Drip rin a Edt cr{c h€ truc l6n tay mdy

b Tinh ma trQn chuydn v!

- Xic dinh

drcrl

asrl

tj

- Tinh cdc ma trQn chuydn v! thtnh phdn (Iit)

[r, -s, o ol [r o o ol [", -s3 o

,'=l; ; I I l, '' =ll I ; :,1,', =1"; "; I

lo o o rJ lo o o rJ Lo o o

- Til dd tinh ma trQn chuydn v! tit gdc ttdn g6c tqa iI| (lit)

Tro =Tro'Tr''Tr'

lo* -crsr t drcrcr+a I

,o _ [ s,c, -srrs -q drsrc,

I

' I r, ca 0 drsr+dr+drl -'

lo o o r j

c Tim vi tri ddu cdng tdc

Nl6n xit: vi trf tldu cong c{c so v6i hQ tga ilQ g6c chinh li vectd:

f drcrcr+a I

rr'=l drsrc,

I

LdSr+dr+dr) Suy ra vi tri bfia ttdu c6ng r6c la:

IJ'

l,=*-0,+d,+5

th6np sd DH (ld

rl

't

) C6u h6i Cho tay m6y c6 2 khAu quay, 1 kh6u tinh fidn nhrrhinh v6.

;- d

Ifr*

=

.iH

a VE c{c hQ trUc 16n tay mdy theo quy tdc D-H

b Tinh ma trfn chuydn vi til cldu c6ng tdc vd gdc tga d0 (tU chqn)

c Tinh v1" tri cria ddu c6ng t6,c khi bidt 0t = rc I 4 , 0z = 0 ,

dt =10(cm).

5

Dr{p 6n a Datcdc hQ-rqc l6n tay m6y

b Tinh ma tran chuydn v!

0

0

dr+ d, I

- Tinh cdc ma trQn chuydn vi thdnh pldn ( ld)

[., 0 sr 0l [t, 0 cz 0l [t o 0

[o o o rJ lo o o r.] [o o o

- Til dd ttnh ma trQn chuydn vi til gdc ddn gtdc toa d0 (\d)

Tro =Tr''Tr''Tr'

DH

1

3

| ,,t, sr c(z (d, + dr)c,c, f

mo I ,,r, -ct s$z (d, + dr)src, I

" =l ,r o -s? dr-(dr+4)s, I

c fim vf tri tldu c6ng ti{c

NhQn xdt: vi tri ddu c6ng tdc so v6i hQ tga cIQ g6c chinh ld

I @r+dr)crc, f pro =l (dr+dr)src, I

lo,-(0,+ d)s,)

Suy ra vi ui ctia ddu c6ng t6c li:

l*=(a,+rolf

l"_

fr=ro,.g+

t

Ciu h6i Cho tay mdy c6 2 kh6u guay, I kh6u tinh tidn nhrlhinh v6

VE c6c h€ trUc l6n tay mdy theo quy tdc D-H

finh ma tr{n ehuydn v! tt tldu c6ng tdc vd gdg tqa d0 (tql chgn)

Tinh v! tri crla ctdu c6ng tdc khi bidt 1t=ft14, 0z=0,

dt =10(cm).

Ddp dn

bdnp thbnp sd DH

b Tinh ina trQ.n chuydn vi

00 00

I dr+d,

01

z=0

3

6

5 Cdu h6i ch" t y m6y c6 2 khdp Quay, 1 khdp dnh tidn nhrf hlnh v6, trong d6 d6

tinh tidn trCn thanh ray theo phrldng Oz cfia hQ tnlc tga d0 g6c'

b

dt;

VE c6c h€ trUc l€n tay m6y theo quy tdc D-H

finh ma trgn chuydn vi til ildu c6ng tdc vd hQ to4 tlQ gdc

Timvi tricria tlduc6ng tdc khi bi€t d, =l}(cm), 0s=0, 0n=rE t4.

\

D6p 6n a D{t c6c hQ trgc l6n tay m6y

4

b Tlnh ma Ean chuydn vi

- Xdc ilinh

- Tfnh cdc 1na trQn chuydn vi tlwnh phdn (1d)

[t o'o ol [t o o d,1 lr, -.s3 o drr,1

',=l: ;l;1,',=ll l; :1,r,=lt "; l'f, 1,

DH 5d

1

I

4

lvu'S

7r -s4 O docof

13 lr, c4 o dotnl

'o=l o o 1 o

t

- Til dd tinh ma n-Qn chuydnv[ til gdc ddn g6c tOa d0 (1'5cl)

Too =Tro 'Trr'Tr'Tot

l rro -ss+ 0 dr+ drcr+ doc,o l

'o =lrro cto o dr+a+drsr+dos,ol

c Tim vi tri ddu c6ng tdc

iia" xit: vitri ddu c6ng tdc so vdi hQ tqa ttQ g6c chinh li vectd:

I ar*drcr+doc* I

lar* o+ drsr+ dos* )

l* = Or+ d'+Vdn

I

| , = o+ro+$a^

1

!- t^a- *,^ *L'*

L

6

6

6 CAU h6i

a.Dit'cdchQ trqc 16n taY mdY

Cho tay rn6y c6 2kh6p quay va I lftop qnn tlen' qrlll uc drlng theo phrldng Ozc$ahQ trqc tqa tlQ g6c'

a VE c6c hQ trUc l€n tay m6y theo quy tic D-H

b Tinh ma trfln chuydn vi tt ddu c6ng tdc vd hO toa dQ gdc'

c Tim vi tri cria clAu c0ng tdc khi bi€t d, =10(cm) ' 0'= 0 ' 0o = 7s I Q

-+ l\

,e

I

E6p 6n

tk

"€ Jil.

I

i

I

i

I

t

''.:

b Tinh ma tran chuydn vi

- Xic dinh bdng thbns sd

- Tinh cdc ma trQn chuydnvi thhnh phln (Id)

ry tl : : :l ^ t; : i ;l m2 t; ;: ; o;":'1

"=lr o r a,l'"=lo I o drl"'=l o o r o l'

[root.j [oooiJ Looo,]

l ro -s4 0 docol

,.r=l"o c4 o o*rl

'4 lo o I o l

lo o o r.l

- Til itd ttnh ma trS.n chuydn vi til gdc ddn g6c tqa it| (1.sir)

Too =Tro.Tr'.fr?fi r' -t

fr* -s34 0 a+drcr+dncu I

ryo lo o -1 o I

'o =lrrn ctc o dr+dr+drsr+4s3, I

lo o o r l

c Tim vitri ddu c6ng tdc

NhQn xit: vi tri adu c6ng tdc so vdi h0 tqa dQ g6c chinh li vectd:

I o* drcr+ dncro I

la, + d, + drs, + as* )

Suy ra vi tri cfia dAu c6ne t6c li:

DH (1.5d

j

rtl

iti

ti

,! I

'|ili ,t:

l

J',

x=a+d3+ d4

!=0

z = dz+rc+f a^

5

7 Cdu h6i Cho tay

a

b

mdy c6 3 khdu quaY nhrr hinh v6'

Idl

I

VE cdc hQ tnlc l€n taY rfl6y.

I'ilnh ma trfnchuydn vi tir cldu c6ng tdc vd tga ttQ gdc

Tynh vi tri cria ddu c6ne t6c khi bidt 0, = tr I 4, 0z = 0, 0t = fi I 4.

L

3

D6p 6n a D{t cdc h0 trqc l€n taY mdY

%Y {x'

I$ lr, A+

lLgJ*.

b Tinh ma trfln chuYdn vi

_tuicanraTni:rrp!,!1t

;: I al

ot

- Tinh cdc ma trQn chuydn vi thdnh phdn (1d)

vdi he tea d0 g6c chlnh li vectd:

d0 (rd)

[r, 0 s, 0l l+ -.e2 0 d,cz

-o_ls, o -q 0l ,,,_ls, c2 0 drs,

tt -l o I o d,l"'-l o o 1 o

[o o o 1] lo o o I

[r, -.r3 0 O*r1

m2 lr, ca 0 4r, I

' l0 0 I 0l

- Tir dd tinh ma tQn chuydn vi til gdc ddn gdc t7a

Tro =Tro.Trt.Tr2

drcrc, + drcrcrrl

drsrcr+4",rr, l

drsr+ Ol"* O,)

| "rru -c't., sr

I

mo I srs, -Jrszr -cr

l =l

' lt, cn o

c Timvi trl ddu c6ng tdc

NhAn xCt: vi ut tldu c6ng t6c so

f drcrcr+ d,rcrcul

Pro =l drsrcr+dSrru I

lasr+asu+a,)

Suy ra vi tri ctia ttdu c6ng tfc li:

1_

*=

,{dr+Jzar1

1-I =

,{ar+Jzar1

z = dr**a,

\

I

CtilltJns s: poNG I{Qc IIIGUIiC R'oEoT coi'{c NGHISF

7;:",:;i,',1.=iilnii,i nii*iou ura tin* uia* p!fusut@

1.

2.

J.

NQi dung bli tor{n dQng hgc ngtlgc

Phr.fdng phdp hinh hoc giii bdi todn ngdQc

Sfi dung ma tr{n chuydn ddi thudn ntrdt ad gi6i bai todn ngrrdc

Ct o t"V *ay mQt khiu dnh tidn, hai kh0u quay' H0 toa d0 gdc dtrQc dflt

nhtl hinh v6:

vd gdc tga dQ:

c6ng tdc li (x"', Y.) vi

nhdt tt ddu c6ng t6c

0 d, + lrc., + trcrrf

0 lrs, +lrsr,

I

r0t

01.]

bidt vi trl c0a tldu

dl

thudn

- sz:

czt 0 0 o2' at Ciu h6i

r

"6 *t tta' thuYdn adi ttrudn nhdt tU

Ca" uid, khdp cAn tim ti d,,0r,0r'Ta ddu c6ng tr{c vd gdc tqa d6:

lrr, -sr 0 dr+lrc, -o lt, c23 o l's'+

t3 =l o o I o

Ta c6 hQ Phrldng trinh:

I er*ot=6oa .

|a, * l.rcosl, + /, cos(0, + 0r) = x"

Ilr rine, + /, sin(o, tor) = !,

Giai he Phrldng trinh tren, ta c6:

( zv"- ,6/, )

0z = acr rtr[-Zr-

)

- ( 2y" -^6/, )

ot =6o0 -acrsinl=jt' )

DLp in

,.:- r':l.S

* .i:

dr t c ' = x" -tr"or( [' orr"in(2Y":-Jit'll-l I zt, )) 2

Ltu j: SV c6 ttrd sfr dUng phuong ph6p hinh hoc dd ei6i

2 CAU h6i

iai

!l

Cho ma trfln chuydn ddi thudq nhdt t[ rIAu c6ng tr{c vd gdc tga dQ:

| ,r", -crsr rr drcrc, + a I

ll

,o _ | s,c, -rrsr -q drsrc,

I

' |

", ca 0 drsr+ dr+ d,

I

Tim gi6 tri c6c bidn khdp bidt vi trf ddu cdng t6c li (+ ,yc,4)

Cho tay mdy c6 hQ toa dQ gdc dugc dAt nhrlhinh v6: 3

D6p 6n c6c bidn ktrdp cdn 0m la 01,d2,03 Ta c6 ma tr{n chuydn odi ttrudn nhdt ru

ddu cOng tdc vd gdc tga ilQ:

ln , -crss sr /rcrcr+a I

,o _ltr", -srsr -q drsrc,

I

' | ", ct 0 drsr+ dr+ d, | i

lo o o I l

Ta c6 hQ phddng ftinh:

| ,*rr+a=xc

f drsrc, = y,

LA* * dr+dr= z"

Giei he phrldng trinh tr6n, ta c6:

o,=acrry(Z-)

\x"-a )

o : d*"or[ '- -""[d, )

sino, )

dz=2"-dr-drsin1,

3

3 CAU h6i Cho tay mdy c6 hQ toa d0 gdc ddqc d{t nhuhinh v6: J

i

L

,i

ii

ii

;t

i1

i

li

il

il

I

ii

il

it

il

ij

ti

l;

I

i

Ire

'i

ffi o>x

cho ma trrn chuyd, f;T^i:rr:,I.ui;:lijiilr

ldc

toa tro:

,o -1r,", -: srsz

!or:,a,)t:: I

I r, 0 -cz d, - (d, + dr)c,

I

Tim gid tri cdc bidn khdp Uidt vi tri Odu c6ng t6c.li (x",yr,z")

Ddp 6n Cr{c bidn khdp cdn tim ld et,e2,d3 Ta c6 ma trQn chuydn Odi thudn nhdt

tt tlAu cdng rdc r,6 gdc tga tlQ:

lrrt, ,sr ctsz (dr+dr)crs, I

,o _l t,r, -q rrsz (dr+ dt)sp,

I

I r, 0 -c2 dr-(dr+ dr)c,

I

Ta c6 hQ phudng trinh:

| (dr+dr)crs2:x"

I ur+dr)srs, = v"

la, -@r+ dr)c, = z,

Giei he phrrong trinh tr6n, ta c6:

'r='*'f(tr)

e,=o*s(@;ffi)

d 5^Z=d'- z" -o^

cos 02

3

4 Cdu h6i Cho tay mdy c6 hQ toa tlQ"gdc cillQc d{t nhrth}nh ve:

, d,

," j ' d'

.t 1-t*,,

y<=

3

.' 16

cho ma trQn chuydn ddi thudn ,ha't tr} ddu c6ng tric vd gdc toa oo;

f n (\ o )n r () r -I\- I

Tim gid

| -12

mo 1",, o

,, =l o l

Lo o

tr! cdc bidn khdp

"12 *l'l

-crz drs,

0 0

bidt v! tri

*3,lol2

I

- (d, + dr)c,, I 0l

1l

tldu c6ng tdc ld (X , y , 0 ) vd 4=0.

Dr{p r{n V6i q:0, cdc bidn khdp cdn tim ld, e2,4.

thudn nhdt tt tldu c6ng tr{c vd gdc tga dQ:

Ta trQn chuydn ddi

Tro

Ta c6 hQ phtrong trinh:

[{ar+a)sr+d, = x"

)

[ -,o +d,)cr=v"

Giei he phrrong trtnh tr6n, ta c6:

oz=acrts(+)

dt=-4 dz

3

5

I

caunai Cho tay mdy c6 hQ tga ilQ g6c nhrlhinh v6:

Cho ma tr{n chuydn

fc

| -31

lo

mOl

lt =1

ts

I "ro L0

Bidt 03 = 0, tim c6c

dl

ddi rt -s:l 0

ctt

0

ridn k

nhfl

,d, bidt

'da

+d,

7+,

tri ,

u c6ngitAc

ctt

losro

uc6

vd gdc tqa tIQ:

I

I

l

ngtdc ld (x ,A,k).

thudn n

'31 o

r-l

n o''

0

khdo b

tUr

lr*

+a

:

vr tl

'rc, + do

ttl

d2

t+

:

:vl

0

drs, + d

I

3

Ddp 6n Vdi q = 0, cdc bidn khdp cAn tim ld, 04,d1 Ta c6 ma trQn chuydn ddi

thudn nhdt trY tldu c6ng tdc vd gdc toa d6:

3

0

s:l 0

o34 dr+drcr+docro

cy 0 dr+a+d3\+dosro

Ta c6 hQ phudng trinh:

ldnco+dr+dr=x"

\roro*a*d,=2"

Giei h0 phrldng trinh tr6n, ta c6

o = acr co"( x" - dr - dt\

v4 - su' vvo[

d4 )

dt = z" - a - (x" - d, - dr)tg9o

6 CAu h6i Cho tay m6y c6 hQ tga d0 g6c nhuhinh v6:

Cho ma

Bi€t.0,

trfn chuydn ddi thudn nhdt til ddu c6ng tic vd gdc tga d0:

l"* -sr+ 0 a+drcr+docro I

,o=lo o -l o

I

| ,r, ctr 0 d, + d, + drs, + dosrn

I

= 0 , tim cdc bidn khdp bidt vi tri crla tlAu c6ng t6c h (x", 0, 4).

3

Ddp 6n Y6i q=A, cdc bidn khdp cdn 0m ld 04,d1 Ta c6.,ma

thudn ntrdt tu ddu c6ng tr{c vd gdc tqa ttQ:

ltro -s,o 0 a+drcr+dncro

I

mo lo o -1 o I

" =1.r, I' ctt o dr+dr+drsr+dnsrl

[o o o 1 ]

ran chuydn tldi

Ta c6 h$ phudng trinh:

ldoco+dr+a=x"

l

[Or, * dr+ d, = z,

Giai he phdong trinh tr6n, ta c6:

( x a-d )

oq=acr*'[=-J

dt = z, - d2 - (x, - d, - a)tg1

3

.:i{i.:i.|',,.:' ,'.;.;:',': j ,'i;;;.]*.

!i * 1

i

k l

&,,.#

7 Cdu h6i c6 3 kh6p quay, mQt khdp tinh tidn vdi c6c bidn khdp

d3, e4 Ma trQn chuydn vi tt} toa dQ tham chidu ddn

0 apt+arcrrl

0 arsr + orrrrl

-l -dr-dn I

0r-J

Tinh c6c bidn k*rdp sao cho robot c6 thd gdp drrQc vflt hinh ch8 nh6t nhr,

hinh vE bidt to4 d0 tam cria vflt li (x", y.) vi phudng cfia v$t hO.p vdi trgc

Ovo m6t s6c a = 45o

+ sl2s4 snc4 -c2s4

*cnsq -coc+-srzsl

0 0

0

0

Cho Robot SCARA

Hn lrrQt ld; 0r, 02,

ddu cOng t6c ld:

|

"rrco

I

I s c

.7 _l -tt-4

A -l I

I

L

I

D6p 6n

- Ma trQn chuydn vi

lcrrco + sr2s4 snc4 - cns4 0 apr + azcrzl

T =lsnco - cnsq - cncc -srzsr 0 ar.tr + azsn

I

Looorl

-Xdc dinh ma trdn hrJdng crla tay kep

4

fn, s., a,1 [-r" ca 9I

R = | n, ', oii,l=l ,, ,'d o

I

ln, ^r- o,) [o o -1]

- Tif ma trQn chuydn vi, ta c6 cdc phrldng tdnh

cnc4 +.s12's4 = -sa

=+ cos(0, + 0, - 0 n)= cos(900 + a)

+ 0, + 0, -00 =900 + a (1)

- Sfi dgng phrfdng phdp hinh hqc, ta tinh dtlgc 02

cosg, '*"' +Y"' -o" -a" - o

2ara,

* 0z =arccosD Q)

- Mic khdc ta c6ng c6 hQ phrldng trinh (d6ng nhdt ma tr6n tinh tidn) farcr+azctz=xc

ja,s,+a2s,.=!"

| -dt-dr=2"

- TU d6 suy ra:

0r = atag(a, + arc2,ct2sr)- atag(x",!") (3)

dt=-dt-2"=-dt,

- Til (1), (2), (3), ta tlnh drrgc 0n

0 o "= A, + 0z -900 - a = 4 + 0z -1350

iI

ii

I Cdu h6i Trinh biy t6m tdt ma tr0n Jacobian vi phrldng phdp tinh? 2

E6p an Ma trQn Jacobian ld ma trfn bidu thi mdi quan hQ gifia vQn tdc tuydn tinh vi

vQn tdc gdc ctia ddu c6ng tdc vdi vQn tdc ctia cdc bidn khdp

Ta c6:

Lr"'l

Trong d6: J,o=[{''l 1u*n, n li sd khiu cfla tay mdy) li ma tr{n Jacobian

" LJ.)'

Ta c6: t" =lt, J,, J,,)

Trong d6:

, I ,,-r(1o -4-,0) : khdp xoay

""'=Lr,-, : khdp tinh tidn

Ta c6: J* =lJ*, J*r J-.]

Trong d6:

, t- z,-,: khdp xoay

't'' =

fo: khdp tlnh tidn

Ydi z,-r ld vebtd clon vi theo trgc z c$a hQ tga d0 thrl i-l

2

2' uau nol Dinh ngtria phrlong trinh dQng trfc hqc? Trinh biy phrlong trinh

Largrange-Euler?

I

D6p 6n DQng lrr c hgc Robot |i phudng trinh vi phdn li6n hQ gifla v{n tdc vi gra tdc

cr{c khiu cfia Robot vdi momen t6c tlQng vio Robot

Ggi q, ti c6c bidn khdp, ta c6 phudng trinh Largrange-Euler:

d aL _aL :,

Trong d6: l

- L =K-U: li hlm Largrange

- K: ttQng n[ng

- U: thdnlng

- ti: momen tdc ilQng vio khdP