1. Introduction 1 2. Mathematical Representation of Robots

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1 1. Introduction Introduction Brief History Types of Robots Technology of Robots Basic Principles in Robotics Notation Symbolic Computation and Numerical Analysis Mathematical Representation of Robots Introduction Position and Orientation of a Rigid Body Some Properties of Rotation Matrices Successive Rotations of a Rigid Body Representation of Orientation by Three Angles Other Representations of Orientation Transformation Between Coordinate Systems Homogeneous Transformation Properties of B[T] A Representation of Joints Rotary Joint Prismatic Joint Screw Joint Cylindrical Joint Spherical Joint Spherical Spherical Joint Pair Other Joints Representation of Links Using Denavit Hartenberg Parameters Link Parameters for Intermediate Links First and Last Links Summary of Link Parameters Link Transformation Matrices 47 Example 2.1: The planar 3R manipulator 48

2 xii Contents Example 2.2: The PUMA 560 manipulator 51 Example 2.3: A SCARA manipulator 53 Example 2.4: The planar four-bar mechanism 55 Example 2.5: A three-dof parallel manipulator 56 Example 2.6: A six-dof parallel manipulator * Homogeneous Coordinates, Lines, Screws, and Twists Kinematics of Serial Manipulators Introduction Degrees of Freedom of a Manipulator Direct Kinematics of Serial Manipulators 71 Example 3.1: The planar 3R manipulator 72 Example 3.2: The PUMA 560 manipulator 72 Example 3.3: A SCARA manipulator Inverse Kinematics of Serial Manipulators 73 Example 3.4: The planar 3R manipulator 74 Example 3.5: The PUMA 560 manipulator Manipulator With Non-intersecting Wrist * Inverse Kinematics of a General 6R Robot Inverse Kinematics for Manipulators With n < Inverse Kinematics of Redundant Manipulators * Solution Methods for Non-linear Equations Kinematics of Parallel Manipulators Introduction Degrees of Freedom Loop-closure Constraint Equations Direct Kinematics of Parallel Manipulators 110 Example 4.1: The planar four-bar mechanism 111 Example 4.2: A three-dof parallel manipulator 114 Example 4.3: A six-dof parallel manipulator * Direct Kinematics of Stewart Gough Platform Mobility of Parallel Manipulators 123 Example 4.4: The planar four-bar mechanism 124 Example 4.5: A three-dof parallel manipulator Inverse Kinematics of Parallel Manipulators 127 Example 4.6: A six-dof hybrid manipulator 129 Example 4.7: The Stewart platform Velocity Analysis and Statics of Manipulators Introduction Linear and Angular Velocities of a Rigid Body 138

3 xiii 5.3 Linear and Angular Velocities of Links in Serial Manipulators 143 Example 5.1: The planar 3R manipulator Serial Manipulator Jacobian Parallel Manipulator Jacobians 152 Example 5.2: The planar four-bar mechanism 155 Example 5.3: A three-dof parallel manipulator Singularities of Serial and Parallel Manipulators 158 Example 5.4: The planar four-bar mechanism 162 Example 5.5: A three-dof parallel manipulator Statics of Serial Manipulators 166 Example 5.6: The planar 3R manipulator Statics of Parallel Manipulators Singularity in Force Domain Resolution of Redundancy at Velocity Level Dynamics of Manipulators Introduction Inertia of a Link The Lagrangian Formulation 187 Example 6.1: Equations of motion of a planar 2R manipulator 192 Example 6.2: Equations of motion of a planar four-bar mechanism Dynamic Equations in Cartesian Space Inverse Dynamics of Manipulators 202 Example 6.3: Inverse dynamics of planar 2R manipulator Simulation of Equations of Motion 204 Example 6.4: Simulation of a planar 2R manipulator 207 Example 6.5: Simulation of a planar four-bar mechanism Recursive Formulations of Dynamics of Manipulators Newton Euler Formulation for Inverse Dynamics * Algorithms for Forward Dynamics * Recursive Algorithms for Parallel Manipulators Trajectory Planning and Generation Introduction Joint Space Schemes 221 Example 7.1: A cubic trajectory Joint Space Schemes With Via Points 226 Example 7.2: A cubic trajectory with a via point 226 Example 7.3: A cubic trajectory with matching velocity and acceleration at a via point 228

4 xiv Contents 7.4 Cartesian Space Schemes Cartesian Straight Line Motion Cartesian Circular Motion Trajectory Planning for Orientation Some Additional Issues in Trajectory Planning Position and Force Control of Manipulators Introduction Feedback Control of a Single-link Manipulator Usefulness of Feedback First-order System Second-order System PID Control of a Single-link Manipulator Digital Control of a Single-link Manipulator PID Control of a Multi-link Manipulator Non-linear Control of Manipulators Time Required to Computethe Model Lack of Knowledge of Model Parameters Simulation and Experimental Results Simulation Results Experimental Results Non-linear Control of Constrained and Parallel Manipulators Cartesian Control of Manipulators Force Control of Manipulators Force Control of a Single Mass Partitioninga Task for Force and Position Control 283 Example 8.1: Peg-in-hole assembly Hybrid Position/Force Controller Stability Analysis of Non-linear Control Schemes Stability Analysis Using Lyapunov s Method 290 Example 8.2: Stability analysis of a single-link manipulator Stability Analysis of PD-and Model-based Control * Advanced Topics in Non-linear Control of Manipulators Modelling and Control of Flexible Manipulators* Introduction Modelling of a Flexible Joint Euler Bernoulli Beam Model Rotating Flexible Link Translating Flexible Link Kinematic Modelling of Multi-link Flexible Manipulators Discretization Methods 319

5 xv Assumed Modes Method FiniteElement Method Comparison of Discretization Methods Equations of Motion of Multi-link Flexible Manipulators Kinetic Energy Potential Energy Symbolic Equations of Motion Control of Flexible Link Manipulators Controllability of Flexible-link Manipulators Model-based Controlfor Trajectory Following End Position Vibration Control A Two-stage Control Algorithm Effect of Uncertainty in Massand Stiffness Numerical Simulation of a Flexible Manipulator Other Topics in Flexible Manipulators Modelling and Analysis of Wheeled Mobile Robots* Introduction Motion of a Single Wheel on Uneven Terrain Model of a Torus-shaped Wheel Representation of Uneven Terrain in R Kinematics of Contact Kinematics of a Single Wheel Dynamics of a Torus-shaped Wheel on Uneven Terrain Kinematic Modelling of a Three-wheeled Mobile Robot Direct and Inverse Kinematics of the 3-DOFWMR Numerical Simulation Results Dynamic Modelling of a Three-wheeled Mobile Robot Equations of Motion of a Three-wheeled Mobile Robot Algorithm for Solving the Equations of Motion Numerical Simulation Results Traversability of a Single Wheel More on Modelling of Wheeled Mobile Robots 407 Index 413

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