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ReferencesChapter 1
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Paul, B., 1963, On the composition of finite rotations, American Mathematical Monthly, 70(8), 859-862.
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Hunt , K. H., 1978, Kinematic Geometry of Mechanisms, Oxford University Press, London U.K.
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Murray, R M., Li, Z., and Sastry, S. S. S., 1994, A Mathematical Introduction to Robotic Manipulation, CRC Press , Boca Raton, Florida.
Nikravesh, P., 1988, Computer-Aided Analysis of Mechanical Systems,Prentice Hall, New Jersey.
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Chapter 4Ball, R S., 1900, A Treatise on the Theory of Screws, Cambridge Uni
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Murray, R M., Li, Z., and Sastry, S. S. S., 1994, A Mathematical Introduction to Robotic Manipulation, CRC Press , Boca Raton, Florida.
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Chapter 8Hunt, K H., 1978, Kinematic Geometry of Mechanisms, Oxford Univer
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Chapter 10Mason, M. T., 2001, Mechanics of Roboti c Manipulation , MIT Press,
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Chapter 12Brady, M., Hollerbach, J . M., Johnson, T . L., Lozano-Prez, T ., and Ma
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Murray, R M., Li, Z., and Sastry, S. S. S., 1994, A Math ematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, Florida.
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Appendix A
Global Frame Triple RotationIn this appendix, the 12 combinat ions of triple rotation about global fixedaxes are presented.
[
cac(3
= C"{sa + cas(3s"(sas"( - caC"{s(3
- c(3sa
caC"{ - sas(3s"(
cas"( + c"(sa s(3
s(3 ]- c(3s"(
c(3C"{
(A.1)
sas"( - cac"(s(3
cac(3
cvso + cas(3s"(
2-QY,,,!Qz ,(3Qx,o.
[
c(3C"{
= s(3- c(3s"(
3-Qz,,,!Qx,(3 Qy,o.
[
cocv - sas(3s"(
= cas"( + C"{sas(3
- c(3sa
- c(3s"(
c(3C"{
s(3
cas"( + C"{sas(3 ]- c(3sa
caC"{ - sas(3s"(
C"{sa + cas(3s"( ]sas"( - cac"(s(3
cac(3
(A.2)
(A .3)
(A.5)
(A.4)sas"( + caC"{s(3 ]
- c"(sa + cas(3s"(
cac(3
- ca s"( + C"{sas(3
cac"( + sas(3s"(
c(3sa
4-Qz ,,,! QY,(3 QX,o.
[
c(3C"{
= c(3s"(- s(3
5-QY,,,!Qx ,(3 Qz,o.
[
caC"{ + sas(3s"( - C"{sa + cas(3s"( C(3S"( ]= ctieo: cac(3 - s(3
- cas"( + C"{sas(3 sas"( + cac"(s(3 c(3C"{
= [ sas"(~~C"{s(3- c"(sa + cas(3s"(
7-QX,,,! QY,(3Q X,o.
= [ S~"(- c"(s(3
- s(3
c(3C"{
c(3s"(
c(3sa ]- cas"( + C"{sas(3
caC"{ + sas(3s"(
(A.6)
(A.7)
676 App endix A. Global Frame Triple Rotation
[
-sas"(+ cac;3q= cas;3
- qsa - cac;3s"(
9-Qz ,,,/QX,(3Q z ,o:
- q s;3c;3
s;3s"(
cas"( +c;3qsa ]sas;3
caq - c;3sas"((A.S)
[
caq - c;3sa s"(= cas"(+ c;3qsa
sas;3
lO-Qx,"/Qz ,(3Qx ,o:
- q sa - cac;3s"(- sa s"(+ cac;3q
cas;3
s;3s"( ]- q s;3
c;3(A.9)
[
c;3 - cas;3= c"(s;3 -sas"(+ cactic»
s;3s"( cvso + cac;3s"(
sas;3 ]- ca s"( - c;3qsacac"( - c;3sas"(
(A.I0)
[
cocv - c;3sa s"(= sas;3
- ca s"( - c;3qsa
I2-Q z,,,/QY,/3Q z, o:
s;3s"(c;3
q s;3
cv so:+ cac;3s"( ]- cas;3
- sa s"(+ coctic»(A.H)
[
- sa s"(+ coctic»= c"(sa + coctie»
- cas;3
- ca s"( - c;3c"(sacaq - c;3sa s"(
sas;3
q s;3 ]s;3s"(c;3
(A.12)
Appendix B
Local Frame Triple RotationIn t his appendix, the 12 combinations of tri ple rotation about local axesare presented.
[
cBc<p= - c'lj;s<p + c<psBs'lj;
s<ps'lj; + c<psBc'lj;(B.1)
2-Ay,1j,Az,oAx,cp
__ [ C_BCS'lj;B s<ps'lj;+ c<psBc'lj; - c<ps'lj;+ sBc'lj;s<p ]cBc<p cBs<p
cBs'lj; - c'lj;s<p + c<psBs'lj; apcib + sBs<ps'lj;
3-Az,1j,Ax,oAy,cp
[
c<pc'ljJ + sBs<ps'lj; cBs'lj; - c'lj;s<p + c<psBs'lj; ]= - c<ps'lj;+ sBc'lj;s<p cBc'lj; s<ps'lj; + c<psBc'lj;
cBs<p - sB cBc<p
4-A z,,pAy,oAx,cp
[
dJc'lj; c<ps'lj;+ sOc'lj;s<p s<ps'ljJ - c<psOc'ljJ ]= - cOs'ljJ c<pc'ljJ - sOs<ps'ljJ c'ljJs<p + c<psBs'ljJ
sO - cOs<p cOc<p
5-Ay,,pAx,oAz,cp
[
c<pc'ljJ - sBs<ps'ljJ c'ljJs<p + c<psBs'lj; - CBS'ljJ]= -cBs<p cBc<p sB
c<ps'ljJ + sBc'ljJ s<p s<ps'lj; - c<psBc'lj; cBc'lj;
[
cOc<p sO - CBS<P]= s<ps'lj; - c<psOc'lj; cBc'lj; c<ps'ljJ + sOc'ljJs<p
c'ljJs<p + c<psBs'ljJ - cOs'lj; c<pc'lj; - sBs<ps'ljJ
7-Ax,,pAy,oAx,cp
[
cB sOs<p - C<PSB]= sBs'ljJ c<pc'lj; - cBs<ps'lj; c'lj;s<p + cOc<ps'ljJ
sBc'ljJ - c<ps'ljJ - cBc'ljJs<p - s<ps'ljJ + cBc<pc'ljJ
(B.2)
(B.3)
(BA)
(B.5)
(B.6)
(B.7)
678 App endix B. Local Frame Triple Rot ation
[
- Si.ps'ljJ + cBci.pc'ljJ sBc'ljJ - Ci.ps'ljJ - cBc'ljJsi.p ]= - ci.psB cB sBsi.p
c'ljJSi.p + cBci.ps'ljJ sBs'ljJ cipcsb - cBsi.ps'ljJ
[
Ci.pc'ljJ - cBsi.ps'ljJ c'ljJSi.p + cBci.ps'ljJ SBS'ljJ ]= - Ci.ps'ljJ - cBc'ljJsi.p - Si.ps'ljJ + cBci.pc'ljJ sBc'ljJ
sBsi.p - ci.psB cB
[
cB ci.psB SBSi.p]= -sBc'ljJ - Si.ps'ljJ + cBci.pc'ljJ Ci.ps'ljJ + cBc'ljJs i.p
sBs'ljJ - c'ljJSi.p - cBci.ps'ljJ cipcsb - cBsi.ps'ljJ
[
Ci.pc'ljJ - cBsi.ps'ljJ sBs'ljJ - c'ljJSi.p - cBci.ps'ljJ ]= sBsi.p cB ci.psB
Ci.ps'ljJ + cBc'ljJsi.p - sBc'ljJ - Si.ps'ljJ + cBci.pc'ljJ
l2-Az ,,pAy,oAz ,<p
[
- Si.ps'ljJ + cBci.pc'ljJ Ci.ps'ljJ + cBc'ljJsi.p - SBC'ljJ]= - c'ljJS i.p - cBci.ps'ljJ cipcsb - cBsi.ps'ljJ sBs'ljJ
ci.psB sBsi.p cB
(B.8)
(B.9)
(B.lO)
(B.ll)
(B.12)
Appendix C
Principal Central Screws TripleCombinationIn this app endix , the six combinat ions of t riple principal central screws arepresented.
l -s(hx ,"(, i) s(hy ,13 ,J) s(hz, a, K)
[
cacj3 - cj3sa= c"(sa + casj3s"( caq - sas j3s"(
sas"( - caqsj3 cas"( + qsasj3o 0
2-s(hy ,13,J) s(hz ,a, K) s(hx ,"(, i)
sj3- cj3s"(cj3q
o
"(px + apzs j3 ]j3py q - apzcj3s"(j3py s"(\apzcj3q
(C.1)
3-s(hz ,a, K) s(hx, "(, i) s(hy, 13 ,J)
[
cac j3sa
- -c~sj3
sj3s"( - cj3qsacocv
cj3s"( + q sasj3o
c"(sj3 + cj3sa s"(- cas"(
cj3q - sas j3s"(o
apzs j3 + "(Px cacj3 ]j3py + "(Px sa
apzcj3 - "(Pxcas j31
(C.2)
r
ca c{3 - sas{3s"( - q sa ca s{3 + c{3sas"(_ ctiso: + cas j3s"( caq sas{3 - cac{3s"(- - qsj3 s"( c{3q
000
4-s(hz ,a ,K) s(hy, (3, J) s(hx, "(, i)
[
cacj3 cas j3s"( - q sa sas"( + caqsj3_ ciie« caq + sas j3s"( q sa sj3 - cas"(- - sj3 cj3s"( cj3q
o 0 0
5-s(hy , 13,J) s(hx, "(, i) s(hz ,a, K)
"(pxca - {3pyqsa ]"(pxsa + j3py cac"(
opz + {3py s"(1
(C.3)
"(Pxcacj3 - j3pysa ]j3py ca + "(Pxcj3sa
apz - "(Px sj31
(C.4)
r
cacj3 + sas j3s"(_ qsa- c{3sa s"( - cas{3
o
cas j3s"( - cj3sacaq
sas j3 + cacj3s"(o
c"(sj3- s "(
c{3qo
"(Px cj3 + apzc"(sj3 ]j3py - apzs,,(
apzcj3q - "(Px sj31
(C.5)
680 Appendix C. Principal Cent ra l Screws Trip le Combination
6-s(hx ,"(, i ) s(hz ,a, k ) s(hy , (3, J)
[
mc(3= s(3s"(+ c(3qsa
c(3sas "( - qs(3a
- sacocvcas"(
a
ms(3q sa s(3 - c(3s"(c(3q + sas(3s"(
a
,,(P X - (3py sa ](3py mq - apzs,,(
apz q +tpy m s"(
(C.6)
Appendix D
Trigonometric FormulaDefinitions in terms of exponentials.
cos z = (D.l)
(D.2)2i
eiz _ e- iz
sinz= - - - -
eiz _ e- i z
tanz = (. .i et Z + e- t z )
ei z = cosz + isinz
e- i z = cos z - isin z
(D.3)
(D.4)
(D.5)
Angle sum and difference.
sin(0:± (3) = sin 0:cos (3 ± cos 0:sin (3
cos(0:± (3) = cos 0:cos (3 =f sin 0:sin (3
( (3)tan 0:± tan (3
tan 0: ± = ----....:...1 =f tan 0:tan (3
( (3)cot 0:cot (3 =f 1
cot 0: ± = ------,,-.:....-:....cot (3 ± cot 0:
(D.6)
(D.7)
(D.8)
(D.9)
Symmetry.
sin( - 0:) = - sin 0:
cos(- 0:) = cos 0:
tan( - 0:) = - tan 0:
(D.lO)
(D.ll)
(D.12)
Multiple angle.
. (2) 2' 2 tan 0:sm 0: = sm 0:cos 0: = 21 + tan 0:
cos(20:) = 2 cos20: - 1 = 1 - 2 sirr' 0: = cos20: - sin20:
2 tan 0:tan(20:) = 2
1 - tan 0:
(2)cot2 0:- 1
cot 0: = 2cot 0:
(D.13)
(D.14)
(D.15)
(D.16)
682 Appendix D. Trigonometric Formula
sin(3a) = -4sin3a +3sina
cos(3a ) = 4cos3 a - 3cosa
(3)- tan" a + 3 tan a
tan a = ---~--3tan2 a + 1
sin(4a) = - 8 sin3 o cos o + 4sinacosa
cos(4a) = 8 cos'' a - 8 cos2 a + 1
(4)- 4 tan' a + 4 tan a
tan a = ---,-------,,--tan? a - 6 tan2 a + 1
sin(5a) = 16sin'' a - 20sin" a + 5sin a
cos(5a) = I fi cos'' a - 20cos 3 a + 5cosa
sin(na) = 2sin((n - l)a) cosa - sin((n - 2)a)
cos(na) = 2cos((n - l )a ) cos a - cos((n - 2)a)
( )tan((n - l )a ) + tan a
tan na = ----'-',..-,..-:..-.;,,..---1 - tan((n - l )a ) ta n a
Half angle.
cos (~) = ±)1+ ~os a
. (a) )1 -cosasin '2 = ± 2
(D.17)
(D.18)
(D.19)
(D.20)
(D.21)
(D.22)
(D.23)
(D.24)
(D.25)
(D.26)
(D.27)
(D.28)
(D.29)
(a) 1 - cosa sin otan - = = = ±
2 sin o 1 + cosa
2tan Q
sin o' = 21 + tan2 -I1 - tarr' Q
cos a = 21 + tan2
Q2
Powers of funct ions.
1 - cosa
1 + cosa(D.30)
(D.31)
(D.32)
. 1sm2 a = 2 (1 - cos(2a))
sin a cos a = ~ sin(2a)
1cos2 a = 2 (1 + cos(2a))
sirr' a = ~ (3s in(a) - sin(3a))
(D.33)
(D .34)
(D.35)
(D.36)
Appendix D. Trigonometric Formula 683
o 1sm2acosa= 4(cosa-3cos(3a)) (Do37)
sin a cos2 a = ~ (sin a + sin(3a)) (Do38)
1cos'' a = 4(cos(3a) + 3cosa)) (Do39)
o 1sin" a = 8 (3 - 4cos(2a) + cos(4a)) (D.40)
1sin3 o coso = 8 (2sin(2a) - sin(4a)) (D.41)
o 1sm2 acos2 a = 8 (1 - cos(4a)) (D.42)
sin a cos" a = ~ (2sin(2a) + sin(4a)) (D.43)
1cos" a = 8 (3 + 4 cos(2a) + cos(4a)) (D.44)
sin'' a = 116
(10sin a - Ssin(3a) + sin(Sa)) (Do4S)
o 1sin" o cos o = 16 (2cosa - 3cos(3a) + cos(Sa)) (D.46)
sin3 acos2 a = 116
(2sina+sin(3a)-sin(Sa)) (D.47)
sirr' o cos'' a = 116
(2 cosa - 3cos(3a) - Scos(Sa)) (D.48)
1sin a cos" a = 16 (2sin a + 3sin(3a) + sin(5a)) (Do49)
1cos'' a = 16 (Iu coso + 5 cos(3a) + cos(Sa)) (DoSO)
tarr' a = 1 - cos(2a) (DoS1)1 + cos(2a)
Products of sin and cos.
1 1cosa cos J3 = "2 cos(a - J3 ) + "2 cos(a + J3 )
o 0 1 1sm o sm J3 = "2 cos(a - J3 ) - "2 cosfo + J3 )
1 1sin a cos J3 = "2 sin(a - J3 ) + "2 sin(a + J3 )
cos a sin J3 = ~ sin(a + J3 ) - ~ sin(a - J3 )
sin(a + J3 )sin(a - J3 ) = cos2 J3 - cos2 a = sin2 a - sin2 J3
(Do52)
(DoS3)
(Do54)
(DoSS)
(DoS6)
684 Appendix D. Trigonometric Formula
cosfo + 13 )cosfo - 13 ) = cos2 13 + sin2 a
Sum of functions .
. ± . 13 2 ' a ±fJ a ±fJsin a sin = sm -2- cos -2-
a+ fJ a- fJcos a + cos 13 = 2 cos -2- cos -2-
13 2 · a +fJ . a- fJcos a -cos = - sm--sm--2 2
sin(a ± 13 )tan o ± tan 13 = 13
cos a cos
sin(fJ ± a )cot a ± cot 13 = . . 13
sm o sm
sin a + sin 13 _ tan~sin a - sin 13 - t an 0:- +13
2
sin o + sin 13 -a + 13---~- = cot ---cos a - cos 13 2
sin o + sin 13 a + 13-----'-:- = tan --cos a + cos 13 2
sin o - sin 13 a - 13-----'--::-=tan--cos a + cos 13 2
Trigonometric relations.
sin2 a - sin2 13 = sin(a + 13 )sin(a - 13 )
cos2 a - cos2 13 = - sin(a + 13 )sin(a - 13 )
(D.57)
(D.58)
(D.59)
(D.60)
(D.61)
(D.62)
(D.63)
(D.64)
(D.65)
(D.66)
(D.67)
(D.68)
Index2R planar manipulat or
cont rol, 652DR t ransformation matrix, 212dynamics, 491, 540equations of motion, 494forward accelerat ion, 439ideal, 491inverse acceleration, 441inverse kinematics, 281, 286,
399inverse velocity, 366, 368Jacobian matrix, 350, 352joint 2 acceleration, 433joint path , 591kinetic energy, 492Lagrange dynamics , 533, 542Lagrangean , 493Newton-Euler dynamics, 516potential energy, 493recursive dynamics, 524time-opt imal cont rol, 630with massive links, 542
3R planar manipulato rDR transformation matrix, 204forward kinematics , 227
4R planar manipulat orstatics, 548
Accelerationangular, 423, 428, 429, 431,
432bias vector, 440body point , 317, 432, 434, 452cent ripetal, 432constant parabola, 593constant path , 580Coriolis, 454discontinuous path, 588
discrete equat ion, 620, 631end-effector, 429forward kinematics, 437, 439gravitational, 532, 538, 549inverse kinematics, 439jump, 573matrix, 414, 423, 434-436recursive, 507, 510sensors, 659tangential, 432
Active transformation, 72Actuator, 7, 12
force and torque, 513, 529,553
optimal torque, 632, 633torque equation, 518, 630
Algorit hmfloating-time, 619, 629inverse kinemati cs, 286LV factorization, 380LV solut ion, 380Newton-Raphson, 398
Angle-axis rotat ion, 106Angular accelerat ion, 423, 431, 432
combination, 428end-effector, 429in terms of Euler parameters,
429, 431in terms of quaternion, 431recursive, 414
Angular momentum2 link manipulator, 462
Angular velocity, 53, 56, 57, 86,299,306
alternative definition, 318combination, 305coordinate transformat ion, 308decomposition, 305
686 Index
elements of matrix, 311in terms of Euler parameters,
310in terms of quaternion, 309in terms of rotation matrix,
307instantaneous, 301instantaneous axis, 302matrix, 300principal matrix, 304recursive, 412, 509
Articulated arm, 8, 231, 267, 357,408
Atan2 function, 272Automorphism, 102Axis-angle rotation, 81, 84, 85, 90,
91, 94
Block diagram , 644Brachisto chrone, 616, 627Bryant angles, 58
Cardanangles, 58frequencies, 58
Cartesianangular velocity, 56end-effector posit ion, 365end-effector velocity, 366manipulat or, 8, 11path, 592
Cent ra l difference, 625Chasles theorem, 154, 166Christoffel operator, 488, 535Co-state variable, 610Cont rol
adapt ive, 649admissible, 618bang-bang , 609, 610characterist ic equation, 646closed-loop, 643command, 643computed force, 651computed torque, 648, 649derivative, 655
desired path , 643error, 643feedback, 644feedback command, 651feedback linearization, 648, 651feedforward command, 651gain-scheduling, 649input , 650integral, 655linear, 649, 654minimum time, 609modified PD , 657open-loop, 643, 650path points, 595PD ,657proportional, 654robots, 13sensing, 657stability of linear, 646time-opt imal, 618, 622, 629,
630, 633time-opt imal description, 618time-opt imal path , 627
Contro ller, 7Coordinate
cylindr ical, 152frame, 17non-Cartesian , 487non-orthogonal, 117parabolic, 487spherical, 153, 332system, 17
Coriolisacceleration, 428, 434effect, 454force, 453
Cycloid, 617
Denavit-Hartenberg, 31meth od, 199, 202, 248nonstand ard method, 223, 283notation, 199parameters, 199, 334, 345, 510,
548
transformation, 208, 212-21 8,220, 222, 243
Differential manifold, 71Differentiating, 312
B-derivative , 312, 314G-derivative, 312, 317second, 320transformation formula, 317
Distal end, 199, 548Dynamics , 421, 507
2R planar manipulator, 516,524
4 bar linkage, 514actuator's force and torque,
529backward Newton-Euler, 522forward Newton-Euler, 529global Newton-Eul er, 511Lagrange , 530Newton-Euler , 511one-link manipulator, 513recursive Newton-Euler, 511,
522
Eartheffect of rotation, 453kinetic energy, 486revolution , 486rot ation , 486rotation effect, 428
Eigenvalue, 87Eigenvector , 87Ellipsoid
energy, 465momentum, 464
End-effector , 6acceleration, 437angular acceleration , 429angular velocity, 363articulated robot , 267configuration vector , 348, 405,
437configuration velocity, 437force, 530frame, 207, 231
Index 687
inverse kinematics, 265kinemati cs, 237link, 199orientation, 271, 364path , 591, 600position, 231position kinematics, 226position vector, 358rotation, 597SCARA position, 149SCARA robot, 240space station manipulator, 243spherical robot , 247time optimal control, 609velocity, 348, 354, 365velocity vector, 348
EnergyEar th kinetic, 486kinetic rigid body, 461kinetic rotational, 458link's kinetic, 531, 537link's potential, 532mechanical, 486point kinetic, 451potential, 489robot kinetic, 531, 538robot potential, 532, 538
Euler-Lexell-Rodriguez formula, 83angles, 18, 48, 51, 53, 107
integrability, 57coordinate frame, 56equation of motion , 457, 460,
461, 466, 467, 513, 523frequencies, 53, 56, 306inverse matrix, 69parameters, 88- 92, 96-98 , 100,
111,309,310rotation matrix, 51, 69theorem, 48, 88
Euler equationbody frame, 460, 467
Euler-Lagrangeequat ion of motion , 614, 615
Eulerian
688 Index
viewpoint , 326
Floating time , 6201 DOF algorithm, 619analytic calculation, 627backward path, 622convergence, 625forward path, 621method,618multi DOF algorithm, 629multiple switching , 633path planning, 627robot control, 629
Force, 449action, 512actuator, 529conservative, 489Coriolis, 454driven , 512driving, 512generalized , 483, 532gravitational vector, 533potential, 489potential field, 485reaction, 512sensors, 660shaking, 516time varying , 454
Forward kinematics, 32Frame
central, 455final, 207goal, 207principal, 457reference, 16special, 206station, 206tool, 207world, 206wrist, 207
Generalizedcoordinate, 480, 483,484, 490force, 482,483,485,487,489,
491, 494, 530
inverse Jacobian, 403Grassmanian, 177Group properties, 70
Hamiltonian, 610Hand , 231Hayati-Roberts notation, 224Helix, 154Homogeneous
compound transformation, 145coordinate, 133, 138direction, 138general transformation, 139,
143inverse transformation, 139,
141, 142, 146position vector , 133scale factor , 133transformation, 131, 134-137,
139, 141
Integrability, 57Inverse kinematics, 32, 265
decoupling technique, 265inverse transformation tech
nique,272iterative technique, 284Pieper technique, 274
Inverted pendulum, 652
Jacobiananalytical, 365elements , 363generating vector, 353, 355,
404geometrical, 365inverse, 287, 403matrix, 285, 287, 290, 292,
348, 352, 355, 357, 361,365, 368, 397, 401, 404,407, 408, 437, 439, 442,534
oflink,531polar manipulator, 349
Jerk
angul ar , 430matrix, 436transformation, 435, 437zero path, 579
Joint , 3acceleration vector, 437act ive, 4coordinate, 4cylindrical, 252inactive, 4orthogonal, 8parallel, 8passive , 4path, 591perp endicular , 8screw, 4variable vector, 348velocity vector, 348, 355
Joint angle, 200Joint distance, 200Joint parameters, 202
Kinematic length, 200Kinematics, 31
acceleration, 423forward , 32, 226forward accelerat ion, 437forward velocity, 348inverse, 32, 265, 272inverse acceleration, 439inverse velocity, 365numerical methods, 377velocity, 345
Kineti c energy, 451Earth, 486link , 537parabolic coordinate, 487rigid body, 461robot , 531, 538rotational body, 458
Kronecker 's delt a , 65, 457, 479
Lagrangedynamics, 530equation, 536
Index 689
equation of motion, 480, 489mechanics , 489multiplier , 617
Lagrange equationexplicit form, 488
Lagrangean , 489, 538robot, 538viewpoint , 326
Lawmotion , 450motion second, 450, 455motion third , 450robotics, 1
Levi-Civita density, 96Lie group, 71Link , 3
angular velocity, 346class 1 and 2, 213class 11 and 12, 218class 3 and 4, 214class 5 and 6, 215class 7 and 8, 216class 9 and 10, 217classification, 219end-effector, 199Euler equat ion, 523kinetic energy, 531Newton-Euler dynamics, 511recursive accelerat ion, 507, 510recursive Newton-Euler dynam-
ics, 522recursive velocity, 509, 510rotational acceleration, 508translational accelerat ion, 508translational velocity, 347velocity, 345
Link length, 200Link offset , 200Link parameters, 202Link twist , 200Location vector, 156, 158LV factorization method, 377, 392
Manipulator2R planar , 491, 533
690 Index
3R planar , 227ar ticulat ed, 205definition, 5inert ia matri x, 532one-link, 490one-link cont rol, 655one-link dynamics, 513PUMA, 204SCARA, 8transformat ion matri x, 267
Mass center, 450, 451, 455Matrix
skew symmet ric, 68, 69, 82,89
Moment , 449action, 512driven, 512driving, 512reaction, 512
Moment of inertiaabout a line, 479ab out a plane, 479about a point , 479characterist ic equation, 477diagonal elements , 477Huygens-Steiner theorem, 471matri x, 468parallel-axes theorem, 469polar , 468principal, 469principal axes, 458principal invariants, 477product , 468pseudo matrix, 469rigid body, 457rotated-axes theorem, 469
Moment of momentum, 450Moment um, 450
angular, 450ellipsoid, 464linear, 450
Motion, 14
Newtonequation of mot ion, 480
Newton equationbody frame, 456global frame, 455Lagrange form, 482rotatin g frame, 453
Newton-Eulerbackward equations, 522equation of motion, 523forward equations, 529global equations, 511recursive equations, 522
Numerical methods, 377analytic inversion, 394Cayley-Hamilton inversion, 395condition number, 388ill-conditioned, 388Jacobian matrix, 404LU factorization, 377LU factorizat ion with pivot-
ing, 383matrix inversion, 390Newton-Ra phson, 398, 400nonlinear equations, 397norm of a matrix, 389partition ing inversion, 393uniqueness of solut ion, 387well-conditioned, 388
Optim al control, 609a linear syste m, 610descript ion, 618first variation, 615Hamiltonian , 610, 613Lagrange equation, 614objective function, 609, 613performance index, 613second variation, 615switching point , 611
Orthogonality condition, 64
Passive transformation, 72Path
Brachistochrone, 627Cartesian, 592constant acceleration, 580
constant angular acceleration,599
cont rol points, 595cubic , 571cycloid, 590har monic, 589higher polynomial, 578jerk zero, 579joint space, 591non-polynomial, 589planning, 592point sequence, 582quadratic, 577quintic, 578rest-to-rest , 573, 574rotati onal, 597splitting, 584to- rest , 573
Pendulumcont rol, 652inverted, 652, 657linear contro l, 655oscillat ing, 484simple, 425, 483spherical, 490
Permutation symbol , 96Phase plane, 611Pieper technique, 274Plucker
angle, 181classification coord inate, 178dist ance, 181line coordinate, 173, 175- 177,
181, 185-187, 247, 248moment , 180ray coordinate , 175, 177reciprocal produ ct , 181screw , 186virtual product , 181
Poinsot 's const ruction, 464Point at infinity, 138Pole, 163Posit ion sensors, 658Positioning, 14Potent ial
Index 691
force, 489Potential energy
robot, 532, 538Proximal end , 199, 548
Quatern ions, 99addition, 99compos it ion rotation, 102flag form, 99inverse rotation, 101multipli cat ion , 99rotation, 100
Rigid bodyacceleration, 431, 508angular momentum, 458angular velocity, 86Euler equation of motion, 461,
466kinematics, 127kinetic energy, 461moment of inertia , 457motion, 127mot ion classification, 167motion compos ition , 131principal rotation matrix , 476rotational kinetics, 457steady rot ation, 462translat ional kinetics, 455velocity, 321, 323
Robotapplicat ion, 13articulated, 8, 231, 238, 267,
357, 361Cartesian , 11classificat ion, 7control, 13, 14cont rol algorithms, 648cylindrical, 11, 259dynamics, 14, 19, 507, 533end-effector path, 600equation of mot ion, 540forward kinematics, 226, 246gravitational vector , 533inertia matrix, 532
692 Index
kinemati cs, 14kinetic energy, 531, 538Lagrange dynamics, 530, 536Lagrange equation, 533Lagrangean, 532, 536link classification, 245modified PD control, 657Newton-Euler dynamics, 511PD control, 657potential energy, 532, 538recursive Newton-Euler dynam-
ics, 522rest position , 200, 203, 231,
235, 239SCARA , 149, 239spherical, 10, 205, 235, 246,
274,355state equat ion, 613stat ics, 546time-opt imal contro l, 613, 629velocity coupling vector, 533
Roboticgeometry, 8histo ry, 1law, 1
Rodriguezrot at ion formula, 83, 84, 89,
92- 95, 101, 106, 114, 128,158, 161, 167, 172, 302,337, 597
vector, 95, 113Roll-pitch-yaw
frequency, 60global angles, 41, 59global rotation matrix, 41, 59
Rotati on, 32, 83about global axis, 33, 38, 40about local axis, 43, 47, 48angle-axis, 106axis-angle, 81, 83-85, 90, 91,
94, 106composit ion, 113decomposition, 113eigenvalue, 87eigenvector, 87
exponential form, 93general, 63infinitesimal, 92local versus global, 61matr ix, 18, 105pole, 326quaternion, 100stanley met hod , 98X-matrix, 33x-matrix, 43Y-matrix, 33y-mat rix, 43Z-matrix, 33z-matrix, 43
Rotational path , 597Rotator, 83, 102
SCARAmanipulator , 8robot , 149, 239
Screw, 154, 157, 166axis, 154centra l, 155, 156, 159, 160,
173, 187, 202, 243, 245,247
combination, 170, 172coordinate, 154decomposition, 172, 173exponential, 171forward kinematics, 243instantaneous, 187intersection, 248inverse, 169, 170, 172left-handed, 155link classificati on, 245location vector, 156motion, 202, 327parameters, 155, 164pitch, 154Plucker coordinate, 186principal, 166, 172, 173reverse central, 156right-h anded, 15, 155special case, 162transformation, 158, 165
twist , 154Second derivative, 320Sensor
accelerat ion, 659position, 658rotary, 658velocity, 659
Sheth not ation , 248Singular configurat ion, 291Spherical coordinate , 153Spinor , 83, 102Spline, 588Stanley method, 98Stark effect, 487Symbols, xi
Ti lt vector, 231Time derivative, 312Top, 53Transformation , 31
active and passive, 71general, 63homogeneous, 131
Transformation matrixderivat ive, 332differential, 336, 337elements, 66velocity, 327
Translati on, 32Triad , 15Trigonometric equation, 271Turn vector , 231Twist vector , 231
Unit system, xiUnit vectors, 16
Index 693
Vectorgravitat ional force, 533velocity coupling, 533
vectorgrav itational force, 537velocity coupling, 536
Vector decomposition, 117Velocity
body point , 452discrete equation, 620, 631end-effector, 348inverse tr ansformation, 330matrix, 436operator matrix, 333prismatic transformat ion, 335revolut e transformation, 335sensors , 659transformation matrix, 327,
329, 331, 333
Work, 451, 454virtual, 483
Work-energy principle, 451Workspace, 11Wrench, 452Wrist , 12-14, 231
decoupling kinematics, 266forward kinematics, 229frame, 207kinematics assembly, 238point , 6, 229, 271position vector, 270spherical, 6, 205, 231, 235, 361transformation matrix 230, ,
267
Zero velocity point , 326