Presents a slightly more abstract (mathematical) formulation of the kinematics, dynamics, and control of robot manipulators. DLC: Robotics.5.2 Screw coordinates for a wrench5.3 Reciprocal screws666 Summary7 Biblgraphy8 E3 Manipulator Kinematics811 Introduction81Forward Kincmatics2.1 Proble statement2.2 The product of exponentia ls formula2.3 Parameterization of manipulators via twists2.4 Manipulator workspace953 Inverse kinematics3.1planar example973.2 Paden-Kahan subproblems93.3 Solving inverse kinematics using subproblems1043.4 General solutions to inverse kinematics problems 1084 The Manipulator Jacobian1154.1 End-effector velocity1154.2 End-effector forces1214.3 Singularities1234.4 Manipulability5 Redundant and Parallel Manipulators5.1 Redundant manipulators1295.2 Parallel manipulators1325.3 Four-bar linkage1355.4 Stewart platform1386 Summary1437 Bibliography1448 Exercises.1464 Robot Dynamics and Control1551 Introduction1552 Lagrange' s Equations1562.1 Basic formulation2.2 Inertial properties of rigid bodies2.3 Example: Dynamics of a two-link planar robot .. 1642.4 Newton-Euler equations for a rigid body1653 Dynamics of Open-Chain Manipulators1683. 1 The Lagrangian for an open-chain robot1683.2 Equations of motion for an open-chain manipulator 1693.3 Robot dynamics and the product of exponentialsformula鲁鲁4 Lyapunov Stability Theory1794.1 Basic definitions4.2 The direct method of Lyapunov1814.3 Thc indirect mcthod of lyapunov1844.4 Examples1854.5Lasalle, s invariance principle5 Position Control and Trajectory Tracking1895.1 Problem description905.2 Computed torque5.3 PD control5.4 Workspace control1956 Control of Constrained manipulators26.1 Dynamics of constrained systems2006.2 Control of constrained manipulato2016.3 Example: A planar manipulator moving in a slot. 203Summary2068 Bibliography2079 Exercises2085 Multifingered Hand Kinematics2111 Introduction to Grasping2112 Grasp statics2142.1 Contact models.2142.2e grasp2183上orce- Closure)5)3.1 Formal definition2233.2 Constructive force-closure conditions4 Grasp Planning2294.1 Bounds on number of required contacts4.2 Constructing force-closure grasps2325 Grasp Constraints5.1 Finger kinematics2345.2 Properties of a multifingered grasp5.3 Fxample: Two SCARA fingers grasping a box2406 Rolling contact Kinematics2426.1 Surface models2436.2 Contact kinematics2486.3 Grasp kinematics with rolling2537 Summary.256Bibliography9 Exercises2596 Hand Dynamics and Control2651 Lagrange's Equations with Constraints2651.1 Pfaffian constraints21. 2 Lagrange multipliers2691.3 Lagrange-d'Alembert formulation1.4 The nature of nonholonomic constraints274e Robot Hand dynamics2762.1 Derivation and properties2762.2 Internal forces2792.3 Other robot systems.2813 Redundant and Nonmanipulable Robot Systems283.1 Dynamics of redundant manipulators2863.22903.3 Example: I wo-fingered Scara grasp04 Kinematics and Statics of Tendon Actuation2934.1 Inelastic tendons2944.2 Elastic tendons294.3 Analysis and control of tendon-driven fingers2985 Control of robot hands3005.1 Extending controllers3005.2 Hierarchical control structurcs3026 Summary3117 Bibliography.313E3147 Nonholonomic Behavior in Robotic Systems3171 Introductio3172 Controllability and Frobenius'Theorem.3212.1 Vector fields and flows3222.2 Lie brackets and Frobenius' theorem3232.3 Nonlinear controllability3283 Examples of Nonholonomic Systems3324 Structure of Nonholonomic SysteIIs4. Classification of nonholonomic distributions344.2 Examples of nonholonomic systems, continued3414.3 Philip lall basis3445 Summary3466 Bibliography3477 Exercises3498 Nonholonomic Motion Planning3551 Introduction355Steering Model Control Systems Using Sinusoids3582.1 First-order controllable systeMs: Brockett's systeIll 3582.2Second-order controllable svstems3612.3 Higher-order systems: chained form systems.... 363 General Methods for Steering3663.1 Fourier techniques3673.2 Conversion to chained form3693.3 Optimal steering of nonholonomic systems3.4 Steering with piecewise COnistant inputs3754 Dynamic Finger Repositioning384.1 Problcm description3824.2 Steering using sinusoids3834.3 Geometric phase algorith3855 Summary3896 Bibliography3907 Exercises3919 Future Prospects3951 Robots in Hazardous Environments3962 Medical Applications for Multifingered Ilands3983 Robots on a small scale: Microrobotics399a Lie Groups and robot Kinematics403Lic groups and robot Kincmatics4031 Differentiable manifolds4031.1 Manifolds and maps4031.2 Tangent spaces and tangent maps4041.3 Cotangent spaces and cotangent maps4051.4 Vector fields4061.5 Differential forms4082 Lie groups4082.1 Definition and exalnples4082.2 The Lie algebra associated with a lie group402.3 Thc exponential map412.4 Canonical coordinates on a Lie group4142.5 Actions of Lie g4153 The Geometry of the Euclidean Group4163.1 Basic properties4163.2 Metric properties of SE(3)4223.3 Volume forms on SE()430B A Mathematica Package for Screw Calculus435Bibliography441Iudex449X11PrefaceIn 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 gencral public as to its sccmingly endless and diversepossibilities. This period has been accompanied by a technological maturation of robots as well, from the simple pick and place and paintingand welding robots, to more sophisticated assembly robots for insertingntegrated circuit chips onto printedbile carts foparts handling and delivery. Several areas of robotic automation havenow become standard on the factory Hoor 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 microlectro-mechanical mechanismsConcurrent with the growth in robotics in the last two decades hasbeell the developinent of courses at Illost Ilajor research universities OIlvarious aspects of robotics. These courses are taught at both the undergraduatc and graduatc levcls in computcr scicncc, clectrical and machanical 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 manipu-latorsGiven 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 kinematicsdynamics, and control of robot manipulators. The current book is anattempt to provide this formulation not just for a single robot but alsofor lllultifingered robot hands, involving Multiple cooperating robots. Itgrew from our efforts to teach a course to a hybrid audience of electricalcnginccrs who did not know much about mcchanisms, computer scicntistswho did not know about control theory, mechanical engineers who weresuspicious of involved explanations of the kinematics and dynamics ofgarden variety open kinematic chains, and mathematicians who were curious, but did not have the time to build up lengthy prerequisites beforebeginning a study of roboticsIt is our premise that abstraction saves time in the long run, in returnfor an initial investment of effort and patience in learning some mathematics. The selection of topics-from kinematics and dynamics of singlerobots, to grasping and manipulation of objects by multifingered robothands, to nonholonomic motion planning-represents an evolution fromthe more basic concepts to the frontiers of the research in the field. Itrepresents what we have used in several versions of the course whichhave been taught between 1990 and 1993 at the University of californiaBerkeley. the Courant Institute of Mathematical Sciences of New YorkUniversity, the California Institute of Technology, and the Hong KongUniversity of Science and Technology(HKUST). We have also presentedparts of this material in short courses at the Universita di roma, theenter for Artificial intellind robotics, bangalore, Indd theNational Taiwan University, Taipei, TaiwanThe material collected here is suitable for advanced courses in roboticsconsisting of seniduate students. At alevel, we cover Chapters 1-4 in a twelve week periodhe course with some discussion of technological and planning issues, aswell as a laboratory. The laboratory consists of experiments involvingon-linc path planning and control of a few industrial robots, and theuse of a simulation environment for off-line programming of robots. Incourses stressing kinematic issues, we often replace material from Chapter4(Robot Dynamics)with selected topics from Chapter 5(MultifingeredIland Kinematics). We have also covered Chapters 5-8 in a ten weekperiod at the graduate level, in a course augmented with other advancedtopics in manipulation or mobile robots.The prerequisites that we assume are a good course in linear algebraat the undergraduate level and some familiarity with signals and systemsA course on control at the undergraduate level is helpful, but not strictlynecessary for following the material. Some amount of mathematical ma-turity is also desirable, although the student who can master the conceptsin Chapter 2 should have no difficulty with the remainder of the bookWe have provided a fair number of exercises after Chapters 2-8 to helpstudents understand some new material and review their understanding ofche chapter. a toolkit of programs written in Mathematica for solving theproblems of Chapters 2 and 3 (and to somc cxtent Chapter 5) have becndeveloped anld are described in Appendix B. We have studiously avoidednumerical exercises in this book: when we have taught the course, wehave adapted numerical exercises from measurements of robots or otherreal"systems available in the laboratories. These vary from one time tothe next and add an element of topicality to the coursee The one large topic in robotic manipulation that we have not coveredin this book is the question of motion planning and collision avoidanceXIVfor robots. In our classroom presentations we have always covered somespects of motion planning for robots for the sake of completeness. Forgraduate classes, we can recommend the recent book of latombe on mo-tion planning as a supplement in this regard. Another omission from thisbook is sensing for robotics. In order to do justice to this material in ourrespective schools, we have always had coinputer vision, tactile sensingand other related topics, such as signal processing, covered in separatecourscsThe contents of our book have been chosen from the point of viewthat they will remain foundational over the next several years in the faceof many new technological innovations and new vistas in robotics. Wehave tried to give a snapshot of some of these vistas in Chapter 9. Inreading this book, we hope that the reader will feel the same excitementchat we do about the technological and social prospects for the field ofrobotics and the elegance of the underlying thncorRichard murrayBerkeley, August 1993Zexiang IShankar Sastr