Handbook of Robotics Chapter 1: Kinematics

1 Handbook of RoboticsChapter 1: KinematicsKen WaldronDepartment of Mechanical EngineeringStanford UniversityStanford, CA 94305, USAJim SchmiedelerDepartment of Mechanical EngineeringThe Ohio State UniversityColumbus, OH 43210, USAS eptember 17, 2007 Contents1 Introduction .. Position and Orientation Representation .. Position and Displacement .. Orientation and Rotation .. 2 Rotation Matrices .. 2 Euler Angles .. 3 Fixed Angles .. 3 Angle-Axis .. 4 Quaternions .. Homogeneous Transformations .. Screw Transformations .. 5 Chasles Theorem .. 6 Rodrigues Equation .. Matrix Exponential Parameterization .. 8 Exponential Coordinates for Rotation .. 8 Exponential Coordinates for Rigid Body motion .. Pl ucker Coordinates .. Joint Kinematics .. Lower Pair Joints .. 10 Revolute .. 10 Prismatic .. 10 Helical .. 10 Cylindrical .. 11 Spherical .. 11 Planar .. Higher Pair Joints.

2 12 Rolling Contact .. Compound Joints .. 12 Universal .. 6-DOF Joint .. Physical Realization .. Holonomic and Nonholonomic Constraints .. Generalized Coordinates .. Geometric Representation .. Workspace .. Forward Kinematics .. Inverse Kinematics .. Closed-Form Solutions .. 17 Algebraic Methods .. 17 Geometric Methods .. Numerical Methods .. 18 Symbolic Elimination Methods .. 18 Continuation Methods .. 18 Iterative Methods .. Forward Instantaneous Kinematics .. Jacobian .. Inverse Instantaneous Kinematics .. Inverse Jacobian .. Static Wrench Transmission .. Conclusions and Further Reading .. 20 List of Initial and final positions of an arbitrary point in a body undergoing a screw theposition of the point relative to the moving frame, which is coincident with the fixed reference framejin its initial the position of the point relative to the fixed frame after the screwdisplacement of the moving body.

3 Schematic of the numbering of bodies and joints in a robotic manipulator, the convention for attachingreference frames to the bodies, and the definitions of the four parameters,ai, i,di, and i, thatlocate one frame relative to another.. Example six-degree-of-freedom serial chain manipulator composed of an articulated arm with nojoint offsets and a spherical wrist.. 15iiiList of Equivalent rotation matrices for various representations of orientation, with abbreviationsc := cos ,s := sin , andv := 1 cos .. Joint model formulas for one-degree-of-freedom lower pair joints, with abbreviationsc i:= cos iands i:= sin i.. Conversions from a rotation matrix to various representations of orientation.. Conversions from angle-axis to unit quaternion representations of orientation and vice versa.. Conversions from a screw transformation to a homogeneous transformation and vice versa, withabbreviationsc := cos ,s := sin , andv := 1 cos.

4 Geometric parameters of the example serial chain manipulator in Figure .. Forward Kinematics of the example serial chain manipulator in Figure , with abbreviationsc i:=cos iands i:= sin i.. Inverse position Kinematics of the articulated arm within the example serial chain manipulator inFigure .. Inverse orientation Kinematics of the spherical wrist within the example serial chain manipulator inFigure , with abbreviationsc i:= cos iands i:= sin i.. Algorithm for computing the columns of the Jacobian from the free modes of the joints.. Joint model formulas for higher-degree-of-freedom lower pair joints, universal joint, rolling contactjoint, and 6-DOF joint, with abbreviationsc i:= cos iands i:= sin i. The Euler angles i, i,and icould be used in place of the unit quaternion ito represent orientation.. 23ivChapter 1 KinematicsKinematics pertains to the motion of bodies in a ro-botic mechanism without regard to the forces/torquesthat cause the motion .

5 Since robotic mechanisms areby their very essence designed for motion , Kinematics isthe most fundamental aspect of robot design, analysis,control, and simulation. The Robotics community hasfocused on efficiently applying different representationsof position and orientation and their derivatives with re-spect to time to solve foundational Kinematics Chapter will present the most useful representa-tions of the position and orientation of a body in space,the Kinematics of the joints most commonly found in ro-botic mechanisms, and a convenient convention for rep-resenting the geometry of robotic mechanisms. Theserepresentational tools will be applied to compute theworkspace, theforwardandinverse Kinematics , theforwardandinverse instantaneous Kinematics , andthestatic wrench transmissionof a robotic mecha-nism. For brevity, the focus will be on algorithms ap-plicable to open-chain goal of this Chapter is to provide the reader withgeneral tools in tabulated form and a broader overviewof algorithms that can be together applied to solve kine-matics problems pertaining to a particular robotic IntroductionUnless explicitly stated otherwise, robotic mechanismsare systems of rigid bodies connected by joints.

6 Theposition and orientation of a rigid body is space are col-lectively termed the pose . Therefore, robot kinematicsdescribes the pose, velocity, acceleration, and all higherorder derivatives of the pose of the bodies that com-prise a mechanism. Since Kinematics does not addressthe forces/torques that induce motion , this Chapter fo-cuses on describing pose and velocity. These descriptionsare foundational elements of dynamics ( Chapter 2), mo-tion planning ( Chapter 5), and motion control (Chapter6) the many possible topologies in which systemsof bodies can be connected, two are of particular impor-tance in Robotics : serial chains and fully parallel mecha-nisms. A serial chain is a system of rigid bodies in whicheach member is connected to two others, except for thefirst and last members that are each connected to onlyone other member. A fully parallel mechanism is one inwhich there are two members that are connected togetherby multiple joints. In practice, each joint is often it-self a serial chain.

7 This Chapter focuses almost exclu-sively on algorithms applicable to serial chains. Parallelmechanisms are dealt with in more detail in Chapter 12 Parallel Mechanisms and Position and OrientationRepresentationSpatial, rigid body Kinematics can be viewed as a com-parative study of different ways of representing the poseof a body. Translations and rotations, referred to in com-bination as rigid body displacements, are also expressedwith these representations. No one approach is optimalfor all purposes, but the advantages of each can be lever-aged appropriately to facilitate the solution of minimum number of coordinates required to lo-cate a body in Euclidean space is six. Many represen-tations of spatial pose employ sets with superabundantcoordinates in which auxiliary relationships exist amongthe coordinates. The number of independent auxiliaryrelationships is the difference between the number of co-ordinates in the set and Chapter and those that follow it make frequent1 Chapter 1.

8 KINEMATICS2use of coordinate reference frames or simply frames .A coordinate reference frameiconsists of an origin, de-notedOi, and a triad of mutually orthogonal basis vec-tors, denoted [ xi yi zi], that are all fixed within a partic-ular body. The pose of a body will always be expressedrelative to some other body, so it can be expressed as thepose of one coordinate frame relative to another. Simi-larly, rigid body displacements can be expressed as dis-placements between two coordinate frames, one of whichmay be referred to as moving , while the other may bereferred to as fixed . This indicates that the observeris located in a stationary position within the fixed ref-erence frame, not that there exists any absolutely Position and DisplacementThe position of the origin of coordinate frameirelativeto coordinate framejcan be denoted by the 3 1 vectorjpi= jpxijpyijpzi .The components of this vector are the Cartesian coor-dinates ofOiin thejframe, which are the projectionsof the vectorjpionto the corresponding axes.

9 The vec-tor components could also be expressed as the sphericalor cylindrical coordinates ofOiin thejframe. Suchrepresentations have advantages for analysis of roboticmechanisms including spherical and cylindrical translation is a displacement in which no point in therigid body remains in its initial position and all straightlines in the rigid body remain parallel to their initialorientations. (The points and lines are not necessarilycontained within the boundaries of the finite rigid body,but rather, any point or line in space can be taken tobe rigidly fixed in a body.) The translation of a body inspace can be represented by the combination of its posi-tions prior to and following the translation. Conversely,the position of a body can be represented as a transla-tion that takes the body from a position in which thecoordinate frame fixed to the body coincides with thefixed coordinate frame to the current position in whichthe two fames are not coincident.

10 Thus, any representa-tion of position can be used to create a representation ofdisplacement, and Orientation and RotationThere is significantly greater breadth in the representa-tion of orientation than in that of position. This sectiondoes not include an exhaustive summary, but focuses onthe representations most commonly applied to rotation is a displacement in which at least one pointof the rigid body remains in its initial position and notall lines in the body remain parallel to their initial orien-tations. For example, a body in a circular orbit rotatesabout an axis through the center of its circular path, andevery point on the axis of rotation is a point in the bodythat remains in its initial position. As in the case of po-sition and translation, any representation of orientationcan be used to create a representation of rotation, MatricesThe orientation of coordinate frameirelative to coordi-nate framejcan be denoted by expressing the basis vec-tors [ xi yi zi] in terms of the basis vectors[ xj yj zj].