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Appendix A

Global Frame Triple RotationIn this appendix, the 12 combinations of triple rotation about global fixedaxes are presented.

1-QX,γQY,βQZ,α

=

⎡⎣ cαcβ −cβsα sβcγsα+ cαsβsγ cαcγ − sαsβsγ −cβsγsαsγ − cαcγsβ cαsγ + cγsαsβ cβcγ

⎤⎦ (A.1)

2-QY,γQZ,βQX,α

=

⎡⎣ cβcγ sαsγ − cαcγsβ cαsγ + cγsαsβsβ cαcβ −cβsα−cβsγ cγsα+ cαsβsγ cαcγ − sαsβsγ

⎤⎦ (A.2)

3-QZ,γQX,βQY,α

=

⎡⎣ cαcγ − sαsβsγ −cβsγ cγsα+ cαsβsγcαsγ + cγsαsβ cβcγ sαsγ − cαcγsβ−cβsα sβ cαcβ

⎤⎦ (A.3)

4-QZ,γQY,βQX,α

=

⎡⎣ cβcγ −cαsγ + cγsαsβ sαsγ + cαcγsβcβsγ cαcγ + sαsβsγ −cγsα+ cαsβsγ−sβ cβsα cαcβ

⎤⎦ (A.4)

5-QY,γQX,βQZ,α

=

⎡⎣ cαcγ + sαsβsγ −cγsα+ cαsβsγ cβsγcβsα cαcβ −sβ

−cαsγ + cγsαsβ sαsγ + cαcγsβ cβcγ

⎤⎦ (A.5)

6-QX,γQZ,βQY,α

=

⎡⎣ cαcβ −sβ cβsαsαsγ + cαcγsβ cβcγ −cαsγ + cγsαsβ−cγsα+ cαsβsγ cβsγ cαcγ + sαsβsγ

⎤⎦ (A.6)

7-QX,γQY,βQX,α

=

⎡⎣ cβ sαsβ cαsβsβsγ cαcγ − cβsαsγ −cγsα− cαcβsγ−cγsβ cαsγ + cβcγsα −sαsγ + cαcβcγ

⎤⎦ (A.7)

864 Appendix A. Global Frame Triple Rotation

8-QY,γQZ,βQY,α

=

⎡⎣ −sαsγ + cαcβcγ −cγsβ cαsγ + cβcγsαcαsβ cβ sαsβ

−cγsα− cαcβsγ sβsγ cαcγ − cβsαsγ

⎤⎦ (A.8)

9-QZ,γQX,βQZ,α

=

⎡⎣ cαcγ − cβsαsγ −cγsα− cαcβsγ sβsγcαsγ + cβcγsα −sαsγ + cαcβcγ −cγsβ

sαsβ cαsβ cβ

⎤⎦ (A.9)

10-QX,γQZ,βQX,α

=

⎡⎣ cβ −cαsβ sαsβcγsβ −sαsγ + cαcβcγ −cαsγ − cβcγsαsβsγ cγsα+ cαcβsγ cαcγ − cβsαsγ

⎤⎦ (A.10)

11-QY,γQX,βQY,α

=

⎡⎣ cαcγ − cβsαsγ sβsγ cγsα+ cαcβsγsαsβ cβ −cαsβ

−cαsγ − cβcγsα cγsβ −sαsγ + cαcβcγ

⎤⎦ (A.11)

12-QZ,γQY,βQZ,α

=

⎡⎣ −sαsγ + cαcβcγ −cαsγ − cβcγsα cγsβcγsα+ cαcβsγ cαcγ − cβsαsγ sβsγ−cαsβ sαsβ cβ

⎤⎦ (A.12)

Appendix B

Local Frame Triple RotationIn this appendix, the 12 combinations of triple rotation about local axesare presented.

1-Ax,ψAy,θAz,ϕ

=

⎡⎣ cθcϕ cθsϕ −sθ−cψsϕ+ cϕsθsψ cϕcψ + sθsϕsψ cθsψsϕsψ + cϕsθcψ −cϕsψ + sθcψsϕ cθcψ

⎤⎦ (B.1)

2-Ay,ψAz,θAx,ϕ

=

⎡⎣ cθcψ sϕsψ + cϕsθcψ −cϕsψ + sθcψsϕ−sθ cθcϕ cθsϕcθsψ −cψsϕ+ cϕsθsψ cϕcψ + sθsϕsψ

⎤⎦ (B.2)

3-Az,ψAx,θAy,ϕ

=

⎡⎣ cϕcψ + sθsϕsψ cθsψ −cψsϕ+ cϕsθsψ−cϕsψ + sθcψsϕ cθcψ sϕsψ + cϕsθcψ

cθsϕ −sθ cθcϕ

⎤⎦ (B.3)

4-Az,ψAy,θAx,ϕ

=

⎡⎣ cθcψ cϕsψ + sθcψsϕ sϕsψ − cϕsθcψ−cθsψ cϕcψ − sθsϕsψ cψsϕ+ cϕsθsψsθ −cθsϕ cθcϕ

⎤⎦ (B.4)

5-Ay,ψAx,θAz,ϕ

=

⎡⎣ cϕcψ − sθsϕsψ cψsϕ+ cϕsθsψ −cθsψ−cθsϕ cθcϕ sθ

cϕsψ + sθcψsϕ sϕsψ − cϕsθcψ cθcψ

⎤⎦ (B.5)

6-Ax,ψAz,θAy,ϕ

=

⎡⎣ cθcϕ sθ −cθsϕsϕsψ − cϕsθcψ cθcψ cϕsψ + sθcψsϕcψsϕ+ cϕsθsψ −cθsψ cϕcψ − sθsϕsψ

⎤⎦ (B.6)

7-Ax,ψAy,θAx,ϕ

=

⎡⎣ cθ sθsϕ −cϕsθsθsψ cϕcψ − cθsϕsψ cψsϕ+ cθcϕsψsθcψ −cϕsψ − cθcψsϕ −sϕsψ + cθcϕcψ

⎤⎦ (B.7)

866 Appendix B. Local Frame Triple Rotation

8-Ay,ψAz,θAy,ϕ

=

⎡⎣ −sϕsψ + cθcϕcψ sθcψ −cϕsψ − cθcψsϕ−cϕsθ cθ sθsϕ

cψsϕ+ cθcϕsψ sθsψ cϕcψ − cθsϕsψ

⎤⎦ (B.8)

9-Az,ψAx,θAz,ϕ

=

⎡⎣ cϕcψ − cθsϕsψ cψsϕ+ cθcϕsψ sθsψ−cϕsψ − cθcψsϕ −sϕsψ + cθcϕcψ sθcψ

sθsϕ −cϕsθ cθ

⎤⎦ (B.9)

10-Ax,ψAz,θAx,ϕ

=

⎡⎣ cθ cϕsθ sθsϕ−sθcψ −sϕsψ + cθcϕcψ cϕsψ + cθcψsϕsθsψ −cψsϕ− cθcϕsψ cϕcψ − cθsϕsψ

⎤⎦ (B.10)

11-Ay,ψAx,θAy,ϕ

=

⎡⎣ cϕcψ − cθsϕsψ sθsψ −cψsϕ− cθcϕsψsθsϕ cθ cϕsθ

cϕsψ + cθcψsϕ −sθcψ −sϕsψ + cθcϕcψ

⎤⎦ (B.11)

12-Az,ψAy,θAz,ϕ

=

⎡⎣ −sϕsψ + cθcϕcψ cϕsψ + cθcψsϕ −sθcψ−cψsϕ− cθcϕsψ cϕcψ − cθsϕsψ sθsψ

cϕsθ sθsϕ cθ

⎤⎦ (B.12)

Appendix C

Principal Central Screws TripleCombinationIn this appendix, the six combinations of triple principal central screws arepresented.

1-s(hX , γ, I) s(hY , β, J) s(hZ , α, K)

=

⎡⎢⎢⎣cαcβ −cβsα sβ γpX + αpZsβ

cγsα+ cαsβsγ cαcγ − sαsβsγ −cβsγ βpY cγ − αpZcβsγsαsγ − cαcγsβ cαsγ + cγsαsβ cβcγ βpY sγ + αpZcβcγ

0 0 0 1

⎤⎥⎥⎦(C.1)

2-s(hY , β, J) s(hZ , α, K) s(hX , γ, I)

=

⎡⎢⎢⎣cαcβ sβsγ − cβcγsα cγsβ + cβsαsγ αpZsβ + γpXcαcβsα cαcγ −cαsγ βpY + γpXsα−cαsβ cβsγ + cγsαsβ cβcγ − sαsβsγ αpZcβ − γpXcαsβ0 0 0 1

⎤⎥⎥⎦(C.2)

3-s(hZ , α, K) s(hX , γ, I) s(hY , β, J)

=

⎡⎢⎢⎣cαcβ − sαsβsγ −cγsα cαsβ + cβsαsγ γpXcα− βpY cγsαcβsα+ cαsβsγ cαcγ sαsβ − cαcβsγ γpXsα+ βpY cαcγ−cγsβ sγ cβcγ αpZ + βpY sγ0 0 0 1

⎤⎥⎥⎦(C.3)

4-s(hZ , α, K) s(hY , β, J) s(hX , γ, I)

=

⎡⎢⎢⎣cαcβ cαsβsγ − cγsα sαsγ + cαcγsβ γpXcαcβ − βpY sαcβsα cαcγ + sαsβsγ cγsαsβ − cαsγ βpY cα+ γpXcβsα−sβ cβsγ cβcγ αpZ − γpXsβ0 0 0 1

⎤⎥⎥⎦(C.4)

5-s(hY , β, J) s(hX , γ, I) s(hZ , α, K)

=

⎡⎢⎢⎣cαcβ + sαsβsγ cαsβsγ − cβsα cγsβ γpXcβ + αpZcγsβ

cγsα cαcγ −sγ βpY − αpZsγcβsαsγ − cαsβ sαsβ + cαcβsγ cβcγ αpZcβcγ − γpXsβ

0 0 0 1

⎤⎥⎥⎦(C.5)

868 Appendix C. Principal Central Screws Triple Combination

6-s(hX , γ, I) s(hZ , α, K) s(hY , β, J)

=

⎡⎢⎢⎣cαcβ −sα cαsβ γpX − βpY sα

sβsγ + cβcγsα cαcγ cγsαsβ − cβsγ βpY cαcγ − αpZsγcβsαsγ − cγsβ cαsγ cβcγ + sαsβsγ αpZcγ + βpY cαsγ

0 0 0 1

⎤⎥⎥⎦(C.6)

Appendix D

Trigonometric FormulaDefinitions in terms of exponentials.

cos z =eiz + e−iz

2(D.1)

sin z =eiz − e−iz

2i(D.2)

tan z =eiz − e−iz

i (eiz + e−iz)(D.3)

eiz = cos z + i sin z (D.4)

e−iz = cos z − i sin z (D.5)

Angle sum and difference.

sin(α± β) = sinα cosβ ± cosα sinβ (D.6)

cos(α± β) = cosα cosβ ∓ sinα sinβ (D.7)

tan(α± β) =tanα± tanβ1∓ tanα tanβ (D.8)

cot(α± β) =cotα cotβ ∓ 1cotβ ± cotα (D.9)

Symmetry.

sin(−α) = − sinα (D.10)

cos(−α) = cosα (D.11)

tan(−α) = − tanα (D.12)

Multiple angle.

sin(2α) = 2 sinα cosα =2 tanα

1 + tan2 α(D.13)

cos(2α) = 2 cos2 α− 1 = 1− 2 sin2 α = cos2 α− sin2 α (D.14)

tan(2α) =2 tanα

1− tan2 α (D.15)

cot(2α) =cot2 α− 12 cotα

(D.16)

870 Appendix D. Trigonometric Formula

sin(3α) = −4 sin3 α+ 3 sinα (D.17)

cos(3α) = 4 cos3 α− 3 cosα (D.18)

tan(3α) =− tan3 α+ 3 tanα−3 tan2 α+ 1 (D.19)

sin(4α) = −8 sin3 α cosα+ 4 sinα cosα (D.20)

cos(4α) = 8 cos4 α− 8 cos2 α+ 1 (D.21)

tan(4α) =−4 tan3 α+ 4 tanαtan4 α− 6 tan2 α+ 1 (D.22)

sin(5α) = 16 sin5 α− 20 sin3 α+ 5 sinα (D.23)

cos(5α) = 16 cos5 α− 20 cos3 α+ 5 cosα (D.24)

sin(nα) = 2 sin((n− 1)α) cosα− sin((n− 2)α) (D.25)

cos(nα) = 2 cos((n− 1)α) cosα− cos((n− 2)α) (D.26)

tan(nα) =tan((n− 1)α) + tanα1− tan((n− 1)α) tanα (D.27)

Half angle.

cos³α2

´= ±

r1 + cosα

2(D.28)

sin³α2

´= ±

r1− cosα

2(D.29)

tan³α2

´=1− cosαsinα

=sinα

1 + cosα= ±

r1− cosα1 + cosα

(D.30)

sinα =2 tan α

2

1 + tan2 α2

(D.31)

cosα =1− tan2 α

2

1 + tan2 α2

(D.32)

Powers of functions.

sin2 α =1

2(1− cos(2α)) (D.33)

sinα cosα =1

2sin(2α) (D.34)

cos2 α =1

2(1 + cos(2α)) (D.35)

sin3 α =1

4(3 sin(α)− sin(3α)) (D.36)

Appendix D. Trigonometric Formula 871

sin2 α cosα =1

4(cosα− 3 cos(3α)) (D.37)

sinα cos2 α =1

4(sinα+ sin(3α)) (D.38)

cos3 α =1

4(cos(3α) + 3 cosα)) (D.39)

sin4 α =1

8(3− 4 cos(2α) + cos(4α)) (D.40)

sin3 α cosα =1

8(2 sin(2α)− sin(4α)) (D.41)

sin2 α cos2 α =1

8(1− cos(4α)) (D.42)

sinα cos3 α =1

8(2 sin(2α) + sin(4α)) (D.43)

cos4 α =1

8(3 + 4 cos(2α) + cos(4α)) (D.44)

sin5 α =1

16(10 sinα− 5 sin(3α) + sin(5α)) (D.45)

sin4 α cosα =1

16(2 cosα− 3 cos(3α) + cos(5α)) (D.46)

sin3 α cos2 α =1

16(2 sinα+ sin(3α)− sin(5α)) (D.47)

sin2 α cos3 α =1

16(2 cosα− 3 cos(3α)− 5 cos(5α)) (D.48)

sinα cos4 α =1

16(2 sinα+ 3 sin(3α) + sin(5α)) (D.49)

cos5 α =1

16(10 cosα+ 5 cos(3α) + cos(5α)) (D.50)

tan2 α =1− cos(2α)1 + cos(2α)

(D.51)

Products of sin and cos.

cosα cosβ =1

2cos(α− β) +

1

2cos(α+ β) (D.52)

sinα sinβ =1

2cos(α− β)− 1

2cos(α+ β) (D.53)

sinα cosβ =1

2sin(α− β) +

1

2sin(α+ β) (D.54)

cosα sinβ =1

2sin(α+ β)− 1

2sin(α− β) (D.55)

sin(α+ β) sin(α− β) = cos2 β − cos2 α = sin2 α− sin2 β (D.56)

872 Appendix D. Trigonometric Formula

cos(α+ β) cos(α− β) = cos2 β + sin2 α (D.57)

Sum of functions.

sinα± sinβ = 2 sin α± β

2cos

α± β

2(D.58)

cosα+ cosβ = 2 cosα+ β

2cos

α− β

2(D.59)

cosα− cosβ = −2 sin α+ β

2sin

α− β

2(D.60)

tanα± tanβ = sin(α± β)

cosα cosβ(D.61)

cotα± cotβ = sin(β ± α)

sinα sinβ(D.62)

sinα+ sinβ

sinα− sinβ =tan α+β

2

tan α−+β2

(D.63)

sinα+ sinβ

cosα− cosβ = cot−α+ β

2(D.64)

sinα+ sinβ

cosα+ cosβ= tan

α+ β

2(D.65)

sinα− sinβcosα+ cosβ

= tanα− β

2(D.66)

Trigonometric relations.

sin2 α− sin2 β = sin(α+ β) sin(α− β) (D.67)

cos2 α− cos2 β = − sin(α+ β) sin(α− β) (D.68)

Index

2R planar manipulatoracceleration analysis, 543assembling, 281control, 837DH transformation matrix, 246dynamics, 622, 695elbow down, 331elbow up, 331equations of motion, 625forward acceleration, 550ideal, 622inverse acceleration, 554inverse kinematics, 331, 359,

505inverse velocity, 466, 468Jacobian matrix, 448, 450joint 2 acceleration, 540joint forces, 660joint path, 750kinematic motion, 332kinetic energy, 623Lagrange dynamics, 675, 696Lagrangean, 624line path, 752Newton-Euler dynamics, 651,

653, 655, 680potential energy, 623recursive dynamics, 664time-optimal control, 812velocity analysis, 413with massive joints, 653, 655,

680with massive links, 696

3R planar manipulatorDH transformation matrix, 238forward kinematics, 260

4R planar manipulatorstatics, 703

Accelerationangular, 529, 534, 536, 538,

539bias vector, 553body point, 399, 539, 541, 584centripetal, 536, 539constant parabola, 755constant path, 738Coriolis, 585discontinuous path, 745discrete equation, 803, 813end-effector, 535forward kinematics, 549, 550gravitational, 671, 692, 703inverse kinematics, 552jump, 731matrix, 530, 541, 548, 566recursive, 557, 560, 641rotational transformation, 530,

535sensors, 844tangential, 536, 539transformation matrix, 541,

542Active transformation, 73Actuator, 7, 13

force and torque, 643, 668,707

optimal torque, 814, 815torque equation, 652, 812

Algorithmfloating-time, 801, 811inverse kinematics, 358LU factorization, 488LU solution, 488Newton-Raphson, 504

Angular acceleration, 529, 538, 539combination, 534

874

end-effector, 535Euler parameters, 536, 538matrix, 530quaternions, 538recursive, 565vector, 530

Angular momentum2 link manipulator, 594

Angular velocity, 56, 59, 60, 98,381

alternative definition, 400combination, 387coordinate transformation, 389decomposition, 387elements of matrix, 393Euler frequencies, 388Euler parameters, 391instantaneous, 383instantaneous axis, 382, 384matrix, 382principal matrix, 385quaternions, 390rate, 382recursive, 440, 559rotation matrix, 388vector, 382

Articulatedarm, 9, 262, 265manipulator, 9, 262, 265, 333,

456Articulated manipulator

equations of motion, 686inverse kinematics, 328, 330,

343inverse velocity, 470Jacobian matrix, 450, 514left shoulder configuration, 349right shoulder configuration,

349Atan2 function, 339Automorphism, 115Axis-angle rotation, 91, 94—96, 103—

105, 107, 120

bac-cab rule, 143

Block diagram, 828Brachistochrone, 798, 809Bryant angles, 61

Cardanangles, 61frequencies, 61

Cartesianangular velocity, 59end-effector position, 464end-effector velocity, 466manipulator, 9, 10path, 754

Central difference, 805Centroid, 407Chasles theorem, 178, 192Christoffel operator, 619, 677Christoffel symbol, 677Co-state variable, 792Control

adaptive, 833admissible, 800bang-bang, 791, 792characteristic equation, 830closed-loop, 827command, 827computed force, 835computed torque, 833derivative, 839desired path, 827error, 828feedback, 828feedback command, 835feedback linearization, 833, 835feedforward command, 835gain, 828gain-scheduling, 833input, 834integral, 839linear, 833, 838minimum time, 791modified PD, 841open-loop, 827, 834path points, 757PD, 841

Index

875

proportional, 839robots, 13sensing, 842stability of linear, 829time-optimal, 801, 804, 811,

812, 815time-optimal description, 800time-optimal path, 809

Controller, 7Coordinate

cylindrical, 176frame, 18non-Cartesian, 618non-orthogonal, 130parabolic, 618spherical, 177, 413system, 18

Coriolisacceleration, 534, 541effect, 585force, 585

Cycloid, 799

Denavit-Hartenberg, 31method, 233, 236, 297nonstandard method, 257, 355notation, 233parameters, 233, 419, 422, 438,

560, 702transformation, 242, 246—252,

254, 256, 292Derivative

coordinate frames, 393transformation formula, 399

Differentialtransformation matrix, 420

Differential manifold, 72Differentiating

B-derivative, 393, 395, 397coordinate frame, 393G-derivative, 393, 399second, 402transformation formula, 399

Direction cosines, 48Distal end, 233, 702

Dynamics, 527, 556, 6412R planar manipulator, 651,

653, 655, 664, 6804 bar linkage, 646actuator’s force and torque,

668backward Newton-Euler, 661forward Newton-Euler, 663global Newton-Euler, 642Lagrange, 669motion, 581Newton-Euler, 641one-link manipulator, 644recursive Newton-Euler, 642,

661robots, 641

Eartheffect of rotation, 585kinetic energy, 617revolution, 617rotation, 617rotation effect, 534

Eigenvaluerotation matrix, 98

Eigenvectorrotation matrix, 98

Ellipsoidenergy, 596momentum, 596

End-effector, 6acceleration, 549angular acceleration, 535angular velocity, 463articulated robot, 333configuration vector, 512, 549configuration velocity, 549force, 663frame, 240inverse kinematics, 325kinematics, 291link, 233orientation, 338, 464path, 749, 763position kinematics, 259

Index

876

position vector, 458rotation, 759SCARA position, 172SCARA robot, 268space station manipulator, 270speed vector, 442, 443spherical robot, 296time optimal control, 791velocity, 454, 465

EnergyEarth kinetic, 617kinetic rigid body, 593kinetic rotational, 589link’s kinetic, 669, 692link’s potential, 671mechanical, 617point kinetic, 583potential, 620robot kinetic, 670, 692robot potential, 671, 692

Euler-Lexell-Rodriguez formula, 93angles, 19, 54, 56, 120integrability, 60

coordinate frame, 59equation of motion, 588, 592,

597, 599, 643, 662frequencies, 56, 59, 388inverse matrix, 71parameters, 102, 103, 105, 110,

111, 113, 124, 391rotation matrix, 54, 71theorem, 51, 102

Euler angles, 52Euler equation

body frame, 592, 599Euler-Lagrange

equation of motion, 796, 798Eulerian viewpoint, 407

Final rotation formula, 101Floating time, 802

1 DOF algorithm, 801analytic calculation, 809backward path, 804

convergence, 807forward path, 803method, 801multi DOF algorithm, 811multiple switching, 815path planning, 809robot control, 811

Force, 581action, 642actuator, 668conservative, 620Coriolis, 585driven, 642driving, 642generalized, 614, 671gravitational vector, 672potential, 620potential field, 616reaction, 642sensors, 844shaking, 648time varying, 586

Forward kinematics, 32Frame

base, 239central, 587final, 240goal, 240neshin, 280principal, 589reference, 17special, 239station, 239takht, 280tool, 240world, 239wrist, 240

Generalizedcoordinate, 611, 614, 615, 621force, 613, 614, 616, 618, 620,

622, 625, 669inverse Jacobian, 509

Grassmanian, 205Group properties, 72

Index

877

Hamiltonian, 792Hayati-Roberts method, 303Helix, 178Homogeneous

compound transformation, 168coordinate, 155, 161direction, 161general transformation, 162,

166inverse transformation, 162,

164, 165, 169position vector, 155scale factor, 155transformation, 154, 156, 158—

162, 165

Integrability, 60Inverse Kinematics

comparison of techniques, 361techniques, 362

Inverse kinematics, 32, 325articulated manipulator, 343decoupling technique, 325Euler angles matrix, 352, 353general formulas, 340inverse transformation tech-

nique, 341iterative algorithm, 358iterative technique, 357nonstandard DH, 355Pieper technique, 343spherical robot, 346

Inverted pendulum, 836

Jacobiananalytical, 464, 465angular, 464displacement matrix, 442elements, 463generating vector, 452, 455,

511geometrical, 464, 465inverse, 359, 509matrix, 358, 359, 362, 364,

442, 443, 450, 454, 456,

460, 461, 465, 469, 504,507, 510, 514, 549, 551,554, 676

of link, 670polar manipulator, 446, 555rotational matrix, 443spherical wrist, 469

Jerkangular, 537matrix, 548rotational transformation, 537transformation, 547, 549transformation matrix, 547zero path, 737

Joint, 3acceleration vector, 549active, 4angle, 235axis, 4coordinate, 4cylindrical, 301distance, 235free, 4inactive, 4orthogonal, 8parallel, 8parameters, 235passive, 4path, 749perpendicular, 8prismatic, 3revolute, 3rotary, 3screw, 4speed vector, 442, 454spherical, 270translatory, 3variable, 4

Kinematic length, 235Kinematics, 31

acceleration, 529assembling, 280direct, 259forward, 32, 233, 259

Index

878

forward acceleration, 549forward velocity, 442inverse, 32, 325, 341inverse acceleration, 552inverse velocity, 465motion, 149numerical methods, 485orientation, 91rigid body, 149rotation, 33surgery, 287velocity, 437

Kinetic energy, 583Earth, 617link, 692parabolic coordinate, 618rigid body, 593robot, 670, 692rotational body, 589

Kronecker delta, 109Kronecker’s delta, 68, 589, 609

Lagrangedynamics, 669equation, 690equation of motion, 611, 620mechanics, 620multiplier, 799

Lagrange equationexplicit form, 619

Lagrangean, 620, 693robot, 693

Lagrangean viewpoint, 407Law

motion, 582motion second, 582, 586motion third, 582robotics, 1

Levi-Civita density, 109Lie group, 72Link, 3

angular velocity, 439class 1 and 2, 247class 11 and 12, 252class 3 and 4, 248

class 5 and 6, 249class 7 and 8, 250class 9 and 10, 251classification, 253end-effector, 233Euler equation, 662kinetic energy, 669length, 235Newton-Euler dynamics, 642offset, 235parameters, 235recursive acceleration, 556, 560recursive Newton-Euler dynam-

ics, 661recursive velocity, 559rotational acceleration, 557translational acceleration, 557translational velocity, 440twist, 235velocity, 437

Location vector, 180, 182LU factorization method, 485, 499

Manipulator2R planar, 622, 6753R planar, 260articulated, 9, 238Cartesian, 9cylindrical, 9definition, 5inertia matrix, 670one-link, 621one-link control, 840one-link dynamics, 644planar polar, 674PUMA, 238SCARA, 9space station, 268, 270spherical, 9transformation matrix, 333

Mass center, 582, 583, 587Matrix

skew symmetric, 70, 71, 92,103

Method

Index

879

Hayati-Roberts, 303non Denavit-Hartenberg, 297parametrically continuous con-

vention, 303Moment, 581

action, 642driven, 642driving, 642reaction, 642

Moment of inertiaabout a line, 610about a plane, 610about a point, 610characteristic equation, 608diagonal elements, 607Huygens-Steiner theorem, 602matrix, 599parallel-axes theorem, 600polar, 599principal, 600principal axes, 589principal invariants, 608product, 599pseudo matrix, 600rigid body, 588rotated-axes theorem, 600

Moment of momentum, 582Momentum, 582

angular, 582ellipsoid, 596translational, 582

Motion, 15

Newtonequation of motion, 611

Newton equationbody frame, 588global frame, 587Lagrange form, 613rotating frame, 585

Newton-Eulerbackward equations, 661equation of motion, 662equations of motion, 642forward equations, 662, 663

global equations, 641recursive equations, 661

Non Denavit-Hartenbergmethods, 297

Non-standardDenavit-Hartenberg method,

257Numerical methods, 485

analytic inversion, 500Cayley-Hamilton inversion, 502condition number, 495ill-conditioned, 494Jacobian matrix, 510LU factorization, 485LU factorization with pivot-

ing, 491matrix inversion, 497Newton-Raphson, 504, 506nonlinear equations, 503norm of a matrix, 496partitioning inversion, 500pivot element, 491uniqueness of solution, 494well-conditioned, 494

Nutation, 52

Object manipulation, 174Optimal control, 791

a linear system, 792description, 800first variation, 797Hamiltonian, 792, 796Lagrange equation, 796objective function, 791, 795performance index, 795second variation, 797switching point, 793

Orthogonality condition, 67

Passive transformation, 73Path

Brachistochrone, 809Cartesian, 754constant acceleration, 738

Index

880

constant angular acceleration,761

control points, 757cubic, 729cycloid, 749harmonic, 748higher polynomial, 735jerk zero, 737joint space, 749non-polynomial, 747planning, 729, 754point sequence, 739quadratic, 734quintic, 736rest-to-rest, 731, 732rotational, 759splitting, 741to-rest, 732

Pendulumcontrol, 836inverted, 836, 842linear control, 840oscillating, 615simple, 532, 614spherical, 621

Permutation symbol, 109Phase plane, 793Pieper technique, 343Plücker

angle, 209axis coordinate, 205classification coordinate, 206distance, 209line coordinate, 201—205, 209,

213—215, 296, 297moment, 208ray coordinate, 203, 205reciprocal product, 209screw, 214virtual product, 209

Poinsot’s construction, 596Point at infinity, 161Polar manipulator

inverse acceleration, 555Pole, 189

Position sensors, 843Positioning, 15Potential

force, 620Potential energy

robot, 671, 692Precession, 52Proximal end, 233, 702

Quaternions, 112, 122addition, 112composition rotation, 115flag form, 112inverse rotation, 114matrix, 123multiplication, 112rotation, 113unit, 124

Rigid bodyacceleration, 538, 558angular momentum, 590angular velocity, 98Euler equation of motion, 592,

597kinematics, 149kinetic energy, 593moment of inertia, 588motion, 149motion classification, 193motion composition, 153principal rotation matrix, 606rotational kinetics, 588steady rotation, 593translational kinetics, 586velocity, 403, 404

Robotapplication, 14articulated, 9, 262, 265, 281,

333, 456, 461Cartesian, 10classification, 8control, 13, 15control algorithms, 833cylindrical, 10, 318

Index

881

dynamics, 15, 20, 556, 641,672, 675

end-effector path, 763equation of motion, 694forward kinematics, 259, 295gravitational vector, 672inertia matrix, 670kinematics, 15kinetic energy, 670, 692Lagrange dynamics, 669, 690Lagrange equation, 672Lagrangean, 671, 678link classification, 294modified PD control, 841Newton-Euler dynamics, 641PD control, 841potential energy, 671, 692recursive Newton-Euler dynam-

ics, 661rest position, 234, 237, 263,

264, 284SCARA, 172, 266spherical, 9, 239, 288, 295, 346,

455state equation, 795statics, 701time-optimal control, 795, 811velocity coupling vector, 672

Roboticgeometry, 8history, 2laws, 1

Rodriguezrotation formula, 93, 95, 103,

104, 106—108, 114, 120,150, 181, 187, 193, 199,384, 421, 759

vector, 109, 127Rodriguez rotation matrix, 109Roll-pitch-yaw

frequency, 62global angles, 44, 62global rotation matrix, 44, 62

Rotation, 32about global axes, 33, 40, 42

about local axes, 46, 50, 51axis-angle, 91, 94—96, 103—105,

107, 120composition, 126decomposition, 126eigenvalue, 98eigenvector, 98exponential form, 106final formula, 101general, 65infinitesimal, 106instantaneous center, 407local versus global, 63matrix, 19, 119pole, 407quaternion, 113Rodriguez formula, 94Rodriguez matrix, 109stanley method, 111Taylor expansion, 124triple global axes, 42X-matrix, 34x-matrix, 47Y-matrix, 34y-matrix, 47Z-matrix, 34z-matrix, 47

Rotational path, 759Rotations

problems, 118Rotator, 94, 116

SCARAmanipulator, 9robot, 172, 266

Screw, 178, 181, 193axis, 178, 408central, 179, 182, 183, 201,

214, 236, 292, 294, 296combination, 198, 200coordinate, 178decomposition, 200, 201exponential, 199forward kinematics, 292instantaneous, 215

Index

882

intersection, 297inverse, 195, 196, 200left-handed, 178link classification, 294location vector, 180motion, 185, 235, 408parameters, 179, 190pitch, 178Plücker coordinate, 214principal, 192, 200, 201reverse central, 179right-handed, 16, 178special case, 188transformation, 181, 191twist, 178

Second derivative, 402Sensor

acceleration, 844position, 843rotary, 843velocity, 843

Sheth notation, 297Singular configuration, 363Singularity, 303Spherical coordinate, 177Spin, 52Spinor, 94, 116Spline, 745Stanley method, 111Stark effect, 618Symbols,

Tilt vector, 275Time derivative, 393Top, 56Torque, 582Transformation, 31

active and passive, 73general, 65homogeneous, 154

Transformation matrixderivative, 417differential, 420, 421elements, 68velocity, 409

Translation, 32Triad, 16Trigonometric equation, 338Turn vector, 275Twist vector, 275

Unit system,Unit vectors, 16

Vectordecomposition, 130gravitational force, 672, 691tilt, 275turn, 275twist, 275velocity coupling, 672, 691

Velocitybody point, 584coefficient matrix, 419discrete equation, 803, 813inverse transformation, 411matrix, 548multiple frames, 405operator matrix, 417prismatic transformation, 419revolute angular matrix, 423revolute transformation, 419rigid body, 403sensors, 843transformation matrix, 409—

412, 417

Work, 583, 586virtual, 614

Work-energy principle, 583Working space, 266Workspace, 13Wrench, 584Wrist, 13—15, 273

classification, 271dead frame, 270decoupling kinematics, 326design, 279Eulerian, 276forward kinematics, 270

xix

xix

Index

883

frame, 240kinematics assembly, 281living frame, 270Pitch-Yaw-Roll, 278point, 6, 270, 271, 337position vector, 336Roll-Pitch-Roll, 276Roll-Pitch-Yaw, 277spherical, 6, 239, 270, 271, 274,

275, 288, 461transformation matrix, 273,

333

Zero velocity point, 407

Index