## Fundamentals of Robotic Mechanical Systems: Theory, Methods, and AlgorithmsModern robotics dates from the late 1960s, when progress in the development of microprocessors made possible the computer control of a multiaxial manipulator. Since then, robotics has evolved to connect with many branches of science and engineering, and to encompass such diverse fields as computer vision, artificial intelligence, and speech recognition.This book deals with robots-such as remote manipulators, multifingered hands, walking machines, flight simulators, and machine tools-that rely on mechanical systems to perform their tasks. It aims to establish the foundations on which the design, control and implementation of the underlying mechanical systems are based. The treatment assumes familiarity with some calculus, linear algebra, and elementary mechanics; however, the elements of rigid-body mechanics and of linear transformations are reviewed in the first chapters, making the presentation self-contained. An extensive set of exercises is included.Topics covered include: kinematics and dynamics of serial manipulators with decoupled architectures; trajectory planning; determination of the angular velocity and angular acceleration of a rigid body from point data; inverse and direct kinematics manipulators; dynamics of general parallel manipulators of the platform type; and the kinematics and dynamics of rolling robots. Since the publication of the previous edition there have been numerous advances in both the applications of robotics (including in laprascopy, haptics, manufacturing, and most notably space exploration) as well as in the theoretical aspects (for example, the proof that Husty's 40th-degree polynomial is indeed minimal - mentioned as an open question in the previous edition). This new edition has been revised and updated throughout to include these new |

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### Contents

An Overview of Robotic Mechanical Systems | xxiii |

12 The General Structure of Robotic Mechanical Systems | 2 |

13 Serial Manipulators | 5 |

14 Parallel Manipulators | 7 |

15 Robotic Hands | 10 |

16 Walking Machines | 12 |

17 Rolling Robots | 14 |

Mathematical Background | 17 |

56 Synthesis of PPO Using Cubic Splines | 201 |

Dynamics of Serial Robotic Manipulators | 209 |

63 Fundamentals of Multibody System Dynamics | 212 |

632 The EulerLagrange Equations of Serial Manipulators | 213 |

633 Kanes Equations | 222 |

Outward Recursions | 223 |

Inward Recursions | 229 |

65 The Natural Orthogonal Complement in Robot Dynamics | 233 |

22 Linear Transformations | 19 |

23 RigidBody Rotations | 24 |

231 The CrossProduct Matrix | 27 |

232 The Rotation Matrix | 29 |

233 The Linear Invariants of a 3 x 3 Matrix | 33 |

234 The Linear Invariants of a Rotation | 34 |

235 Examples | 36 |

236 The EulerRodrigues Parameters | 42 |

24 Composition of Reflections and Rotations | 46 |

25 Coordinate Transformations and Homogeneous Coordinates | 47 |

251 Coordinate Transformations Between Frames with a Common Origin | 48 |

252 Coordinate Transformation with Origin Shift | 51 |

253 Homogeneous Coordinates | 53 |

26 Similarity Transformations | 57 |

27 Invariance Concepts | 62 |

271 Applications to Redundant Sensing | 65 |

Fundamentals of RigidBody Mechanics | 70 |

32 General RigidBody Motion and Its Associated Screw | 71 |

321 The Screw of a RigidBody Motion | 73 |

322 The Plucker Coordinates of a Line | 75 |

323 The Pose of a Rigid Body | 79 |

33 Rotation of a Rigid Body About a Fixed Point | 82 |

34 General Instantaneous Motion of a Rigid Body | 83 |

341 The Instant Screw of a RigidBody Motion | 84 |

342 The Twist of a Rigid Body | 87 |

35 Acceleration Analysis of RigidBody Motions | 90 |

36 RigidBody Motion Referred to Moving Coordinate Axes | 92 |

37 Static Analysis of Rigid Bodies | 94 |

38 Dynamics of Rigid Bodies | 98 |

Kinetostatics of Simple Robotic Manipulators | 103 |

42 The DenavitHartenberg Notation | 104 |

43 The Kinematics of SixRevolute Manipulators | 112 |

44 The IKP of Decoupled Manipulators | 116 |

441 The Positioning Problem | 117 |

442 The Orientation Problem | 132 |

45 Velocity Analysis of Serial Manipulators | 137 |

451 Jacobian Evaluation | 144 |

452 Singularity Analysis of Decoupled Manipulators | 149 |

453 Manipulator Workspace | 151 |

46 Acceleration Analysis of Serial Manipulators | 155 |

47 Static Analysis of Serial Manipulators | 159 |

481 Displacement Analysis | 162 |

482 Velocity Analysis | 164 |

483 Acceleration Analysis | 167 |

484 Static Analysis | 169 |

49 Kinetostatic Performance Indices | 170 |

491 Positioning Manipulators | 175 |

492 Orienting Manipulators | 178 |

493 Positioning and Orienting Manipulators | 179 |

Trajectory Planning PickandPlace Operations | 188 |

52 Background on PPO | 189 |

53 Polynomial Interpolation | 191 |

532 A 4567 Interpolating Polynomial | 195 |

54 Cycloidal Motion | 198 |

55 Trajectories with Via Poses | 200 |

651 Derivation of Constraint Equations and TwistShape Relations | 239 |

652 Noninertial Base Link | 243 |

661 Planar Manipulators | 247 |

662 Algorithm Complexity | 260 |

663 Simulation | 264 |

67 Incorporation of Gravity Into the Dynamics Equations | 267 |

68 The Modeling of Dissipative Forces | 268 |

Special Topics in RigidBody Kinematics | 272 |

72 Computation of Angular Velocity from PointVelocity Data | 273 |

73 Computation of Angular Acceleration from PointAcceleration Data | 279 |

Kinematics of Complex Robotic Mechanical Systems | 286 |

82 The IKP of General SixRevolute Manipulators | 287 |

821 Preliminaries | 288 |

822 The BivariateEquation Approach | 301 |

823 The UnivariatePolynomial Approach | 303 |

824 Numerical Conditioning of the Solutions | 312 |

825 Computation of the Remaining Joint Angles | 313 |

826 Examples | 316 |

83 Kinematics of Parallel Manipulators | 321 |

831 Velocity and Acceleration Analyses of Parallel Manipulators | 336 |

84 Multifingered Hands | 342 |

85 Walking Machines | 347 |

86 Rolling Robots | 351 |

862 Robots with Omnidirectional Wheels | 357 |

Trajectory Planning ContinuousPath Operations | 361 |

92 Curve Geometry | 362 |

93 Parametric Path Representation | 369 |

94 Parametric Splines in Trajectory Planning | 382 |

95 ContinuousPath Tracking | 388 |

Dynamics of Complex Robotic Mechanical Systems | 400 |

102 Classification of Robotic Mechanical Systems with Regard to Dynamics | 401 |

103 The Structure of the Dynamics Models of Holonomic Systems | 402 |

104 Dynamics of Parallel Manipulators | 405 |

105 Dynamics of Rolling Robots | 416 |

1052 Robots with Omnidirectional Wheels | 426 |

Kinematics of Rotations A Summary | 436 |

The Numerical Solution of Linear Algebraic Systems | 443 |

B1 The Overdetermined Case | 445 |

B11 The Numerical Solution of an Overdetermined System of Linear Equations | 446 |

B2 The Underdetermined Case | 450 |

B21 The Numerical Solution of an Underdetermined System of Linear Equations | 451 |

Exercises | 453 |

2 Mathematical Background | 456 |

3 Fundamentals of RigidBody Mechanics | 464 |

4 Kinetostatics of Simple Robotic Manipulators | 470 |

PickandPlace Operations | 477 |

6 Dynamics of Serial Robotic Manipulators | 480 |

7 Special Topics on RigidBody Kinematics | 486 |

8 Kinematics of Complex Robotic Mechanical Systems | 489 |

ContinuousPath Operations | 493 |

10 Dynamics of Complex Robotic Mechanical Systems | 497 |

References | 500 |

513 | |

### Common terms and phrases

3-dimensional vector acceleration actuated joint algorithm angle angular velocity apparent array axes axis calculated Cartesian coefficients components computed condition number configuration constraints coordinate frame Darboux vector decoupled manipulators defined denoted derived determined displayed dynamics eigenvalues end-effector entries Euler-Lagrange equations Euler-Rodrigues parameters expression FIGURE foregoing equation function Furthermore given hence identical intersection invariants inverse kinematics isotropic Jacobian matrix joint rates joint variables joint-rate kinematic pairs manipulator of Fig mass center Moreover motion multiplications namely Newton-Euler nullspace obtain orientation orthogonal orthogonal matrix parallel manipulators planar plane platform Plucker polynomial pose position vector prismatic pair problem readily relation respect revolute right-hand side rigid body robotic manipulator robotic mechanical systems rolling robot rotation matrix scalar equations Section serial manipulators shown in Fig singular solution solve spherical spline Subsection Theorem thereby time-derivatives tion torques trajectory twist unit vector values vanishes wheels workspace wrench wrist zero