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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