A Mathematical Introduction to Robotic Manipulation

1 A Mathematical Introduction toRobotic ManipulationRichard M. MurrayCalifornia Institute of TechnologyZexiang LiHong Kong University of Science and TechnologyS. Shankar SastryUniversity of California, Berkeleyc 1994, CRC PressAll rights reservedThis electronic edition is available ~ editions may be purchased from CRC Press, manuscript is for personal use only and may not be reproduced, inwhole or in part, without written consent from the RuthAnne (RMM)To Jianghua (ZXL)In memory of my father (SSS)viContentsContentsviiPrefacexiiiAck nowledgementsxvii1 Introduction11 Brief History .. 12 Multifingered Hands and Dextrous Manipulation .. 83 Outline of the Book .. Manipulation using single robots .. Coordinated Manipulation using multifingered robothands .. Nonholonomic behavior in Robotic systems.

2 164 Bibliography .. 182 Rigid Body Motion191 Rigid Body Transformations .. 202 Rotational Motion inR3.. Properties of rotation matrices .. Exponential coordinates for rotation .. Other representations .. 313 Rigid Motion inR3.. Homogeneous representation .. Exponential coordinates for rigid motion and twists Screws: a geometric description of twists .. 454 Velocity of a Rigid Body .. Rotational velocity .. Rigid body velocity .. Velocity of a screw motion .. Coordinate transformations .. 585 Wrenches and Reciprocal Screws .. Wrenches .. Screw coordinates for a wrench .. Reciprocal screws .. 666 Summary .. 707 Bibliography .. 728 Exercises .. 733 Manipulator Kinematics811 Introduction .

3 812 Forward Kinematics .. Problem statement .. The product of exponentials formula .. Parameterization of manipulators via twists .. Manipulator workspace .. 953 Inverse Kinematics .. A planar example .. Paden-Kahan subproblems .. Solving inverse kinematics using subproblems .. General solutions to inverse kinematics problems . 1084 The Manipulator Jacobian .. End-effector velocity .. End-effector forces .. Singularities .. Manipulability .. 1275 Redundant and Parallel Manipulators .. Redundant manipulators .. Parallel manipulators .. Four-bar linkage .. Stewart platform .. 1386 Summary .. 1437 Bibliography .. 1448 Exercises .. 1464 Robot Dynamics and Control1551 Introduction .. 1552 Lagrange s Equations.

4 Basic formulation .. Inertial properties of rigid bodies .. Example: Dynamics of a two-link planar robot .. Newton-Euler equations for a rigid body .. 1653 Dynamics of Open-Chain Manipulators .. The Lagrangian for an open-chain robot .. Equations of motion for an open-chain manipulator Robot dynamics and the product of exponentialsformula .. 1754 Lyapunov Stability Theory .. Basic definitions .. The direct method of Lyapunov .. The indirect method of Lyapunov .. Examples .. Lasalle s invariance principle .. 1885 Position Control and Trajectory Tracking .. Problem description .. Computed torque .. PD control .. Workspace control .. 1956 Control of Constrained Manipulators .. Dynamics of constrained systems.

5 Control of constrained manipulators .. Example: A planar manipulator moving in a slot . 2037 Summary .. 2068 Bibliography .. 2079 Exercises .. 2085 Multifingered Hand Kinematics2111 Introduction to Grasping .. 2112 Grasp Statics .. Contact models .. The grasp map .. 2183 Force-Closure .. Formal definition .. Constructive force-closure conditions .. 2244 Grasp Planning .. Bounds on number of required contacts .. Constructing force-closure grasps .. 2325 Grasp Constraints .. Finger kinematics .. Properties of a multifingered grasp .. Example: Two SCARA fingers grasping a box .. 2406 Rolling Contact Kinematics .. Surface models .. Contact kinematics .. Grasp kinematics with rolling .. 2537 Summary .. 2568 Bibliography.

6 2579 Exercises .. 259ix6 Hand Dynamics and Control2651 Lagrange s Equations with Constraints .. Pfaffian constraints .. Lagrange multipliers .. Lagrange-d Alembert formulation .. The nature of nonholonomic constraints .. 2742 Robot Hand Dynamics .. Derivation and properties .. Internal forces .. Other robot systems .. 2813 Redundant and Nonmanipulable Robot Systems .. Dynamics of redundant manipulators .. Nonmanipulable grasps .. Example: Two-fingered SCARA grasp .. 2914 Kinematics and Statics of Tendon Actuation .. Inelastic tendons .. Elastic tendons .. Analysis and control of tendon-driven fingers .. 2985 Control of Robot Hands .. Extending controllers .. Hierarchical control structures .. 3026 Summary.

7 3117 Bibliography .. 3138 Exercises .. 3147 Nonholonomic Behavior in Robotic Systems3171 Introduction .. 3172 Controllability and Frobenius Theorem .. Vector fields and flows .. Lie brackets and Frobenius theorem .. Nonlinear controllability .. 3283 Examples of Nonholonomic Systems .. 3324 Structure of Nonholonomic Systems .. Classification of nonholonomic distributions .. Examples of nonholonomic systems, continued .. Philip Hall basis .. 3445 Summary .. 3466 Bibliography .. 3477 Exercises .. 3498 Nonholonomic Motion Planning3551 Introduction .. 3552 Steering Model Control Systems Using Sinusoids .. First-order controllable systems: Brockett s system Second-order controllable systems .. Higher-order systems: chained form systems.

8 3633 General Methods for Steering .. Fourier techniques .. Conversion to chained form .. Optimal steering of nonholonomic systems .. Steering with piecewise constant inputs .. 3754 Dynamic Finger Repositioning .. Problem description .. Steering using sinusoids .. Geometric phase algorithm .. 3855 Summary .. 3896 Bibliography .. 3907 Exercises .. 3919 Future Prospects3951 Robots in Hazardous Environments .. 3962 Medical Applications for Multifingered Hands .. 3983 Robots on a Small Scale: Microrobotics .. 399A Lie Groups and Robot Kinematics403 Lie Groups and Robot Kinematics4031 Differentiable Manifolds .. Manifolds and maps .. Tangent spaces and tangent maps .. Cotangent spaces and cotangent maps .. Vector fields .. Differential forms.

9 4082 Lie Groups .. Definition and examples .. The Lie algebra associated with a Lie group .. The exponential map .. Canonical coordinates on a Lie group .. Actions of Lie groups .. 4153 The Geometry of the Euclidean Group .. Basic properties .. Metric properties ofSE(3) .. Volume forms onSE(3) .. 430B A Mathematica Package for Screw Calculus435 Bibliography441 Index449xixiiPrefaceIn the last two decades, there has been a tremendous surge of activityin robotics, both at in terms of research and in terms of capturing theimagination of the general public as to its seemingly endless and diversepossibilities. This period has been accompanied by a technological mat-uration of robots as well, from the simple pick and place and paintingand welding robots, to more sophisticated assembly robots for insertingintegrated circuit chips onto printed circuit boards, to mobile carts forparts handling and delivery.

10 Several areas of Robotic automation havenow become standard on the factory floor and, as of the writing ofthis book, the field is on the verge of a new explosion to areas of growthinvolving hazardous environments, minimally invasive surgery, and microelectro-mechanical with the growth in robotics in the last two decades hasbeen the development of courses at most major research universities onvarious aspects of robotics. These courses are taught at both the under-graduate and graduate levels in computer science, electrical and mechan-ical engineering , and mathematics departments, with different emphasesdepending on the background of the students. A number of excellenttextbooks have grown out of these courses, covering various topics inkinematics, dynamics, control, sensing, and planning for robot the state of maturity of the subject and the vast diversity of stu-dents who study this material, we felt the need for a book which presentsa slightly more abstract ( Mathematical ) formulation of the kinematics,dynamics, and control of robot manipulators.